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

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.


Introduction
The PT -symmetric scalar theory gφ2 (iφ) ǫ was recently studied within the framework of the logarithmic expansion, that is an expansion in powers of ǫ.The Green's functions G n were calculated at first order in ǫ, and it was proposed to use this expansion to implement a systematic renormalization of the theory [1].
Of great physical interest is the case ǫ = 1, the igφ3 model, introduced some time ago by Fisher [2] to study the density of the Lee-Yang zeros of the partition function Z[h] on the imaginary axis of h (h is the magnetic field).Its renormalization properties were investigated in [3,4], where the critical exponents were calculated.This model is the field theoretic analogue of the PT -symmetric quantum-mechanical theory described by the Hamiltonian H = p 2 + ix 3 , whose eigenvalues have been rigorously shown to be all real [5].While in the case of PT -symmetric quantum mechanics, however, the boundary conditions on the Schrodinger equation are imposed in complex Stokes sectors [6], the search for the Stokes wedges for the integration over infinitely many complex field variables is an insurmountable task.The advancement that the expansion in ǫ seems to bring is that, for sufficiently small values of ǫ, it allows to perform the functional integration for the calculation of the Green's functions along the real-φ axis, rather than in the complex-φ domain.
Clearly, for studying and understanding the potentialities of this expansion, the higher orders need to be considered.In this respect, we note that even for the corresponding ordinary theory gφ 2 (φ 2 ) ǫ , to which the logarithmic expansion was previously applied [7,8], only the O(ǫ) was systematically calculated, while only partial results exist for the O(ǫ 2 ) [9,10].Now, since the logarithmic expansion and the non-hermitian gφ 2 (iφ) ǫ theory have each their own peculiarities, in order to better investigate the properties both of the expansion and of the PT -symmetric theory itself, we decided to disentangle the two investigations, splitting our analysis in two different works.The present paper is the second one of this series.
In the first paper of this series [11] we studied the higher orders in the expansion of the ordinary theory gφ 2 (φ 2 ) ǫ .We found that at any finite order in ǫ (for dimension d ≥ 2) the theory is non-interacting, i.e. all the Green's function G n with n ≥ 4 vanish, and concluded that physically sensible results can be obtained only resorting to resummations.However, resumming leading and subleading contributions to the G n , we found non-vanishing but quite trivial results.In the particular case of ǫ = 2, the ordinary gφ 4 theory, these resummations lead to the first and second order results of the weak-coupling expansion.
The goal of the present work is to study the Green's functions of the PT -symmetric gφ 2 (iφ) ǫ theory at higher orders in ǫ (the O(ǫ) was considered in [1]).For this analysis, we will take great advantage of the techniques that we developed in our previous paper [11].Interestingly we will see that the results for the PT -symmetric theory closely parallel those that we found for the ordinary theory.
We begin by calculating the Green's functions at O(ǫ 2 ), giving a closed form for all the G n in terms of hypergeometric functions, and find that up to this order the theory is non-interacting.We then move to consider higher orders.The result does not change: at each finite order in ǫ, the theory is non-interacting for any dimension d ≥ 2. We then consider resummations of the Green's functions, and find that, as it was the case for the corresponding ordinary theory, the results that we obtain are quite deceptive.Specifying to the igφ 3 model, we see that the resummations simply give the lowest orders of the weak-coupling expansion.
The rest of the paper is organized as follows.In Section 2 we begin by setting up the tools for our analysis, and review some previous results.We also analyse the UV behaviour of the O(ǫ) contributions to the G n .In Section 3 we move to the O(ǫ 2 ), obtaining general expressions for all the Green's functions G n .We study the UV behaviour of the G n , and then extend the analysis to higher orders in Section 4. Section 5 is devoted to the resummation of the UV-leading contributions at each order in ǫ, while in Section 6 we consider the resummation of subleading terms.Section 7 is for the conclusions.

Green's functions at O(ǫ) and their UV behaviour
In the present section, after sketching the steps for the calculation of the G n at each order in ǫ, we review the first order calculations, thus obtaining the results of [1], with some generalizations detailed below.We then analyse the UV behaviour of the Green's functions, showing that at this order the theory is non-interacting.We also introduce "effective vertices" that will play a very important role for the analysis of the theory at higher orders.
Let us consider the PT -symmetric (Euclidean) lagrangian in d dimensions (ǫ ≥ 0): that is an extension of the lagrangian considered in [1], where we introduce an explicit mass term m 2 φ 2 , and use dimensionful parameters, with g being the coupling constant and µ the 't Hooft scale.Expanding iµ 1−d/2 φ ǫ in the interaction lagrangian in powers of ǫ, and including the ǫ 0 term in the "free lagrangian" (see below), (1) can be written as with L 0 and L p given by As φ is real, the complex logarithm is log iµ , so L p can be splitted as The expansion of the Green's functions in the parameter ǫ is obtained by treating L int as a perturbation, and expanding as usual the exponential e −S int in the path integral that defines the G n .We will consider only connected Green's functions.
From (5) we see that all the "interaction terms" L pk contain factors of the kinds φ 2 log k (µ 2−d φ 2 ) and iπ|φ|/φ.Therefore, in addition to the terms that appear in the corresponding ordinary theory gφ 2 (φ 2 ) ǫ considered in [11], in the present case also powers of |φ|/φ appear.The calculation of the different orders in ǫ of the connected Green's functions can be done as sketched below [1].The powers of log φ 2 are replaced by the formal identity (that is an adaptation of the replica trick [12]) and the factors φ|φ| are written with the help of the integral formula The variables N i are initially treated as positive integer numbers, so that from both ( 6) and ( 7) the functional integrals for the calculation of the G n are traced back to an application of the Wick's theorem.The results are then analytically extended to real values of the N i 's.

Green's functions at O(ǫ). Effective vertices.
To put at work the above machinery, we begin by calculating the O(ǫ) contribution to the connected Green's functions, thus recovering results similar to those of [1], but relative to the lagrangian (1).From (2) we have (we need only the free partition function Z 0 as we consider connected G n ): From (5) we see that Since L 1 0 is even in φ while L 1 1 is odd, they contribute to even and odd G n respectively.We treat them separately.Odd Green's functions.The O(ǫ) contributions to the odd Green's functions come from L 1 1 .From ( 5) and ( 7) we then have: For r < n−3 2 the functional integral vanishes, as we cannot draw connected diagrams, while for r where B n (r) is the combinatorial factor coming from the different contactions and The loop integral ∆(0) diverges for d ≥ 2, and for the purposes of our later analysis we will regularize this divergence with the help of a physical momentum cut-off, that is Due to the presence of the falling factorial (r + 1) n−1

2
, the coefficient B n (r) vanishes for r < n−3 2 , so that (10) holds true for all positive integers r.Therefore, the O(ǫ) contribution (9) to the odd Green's functions can be written as where, following our previous work on the corresponding ordinary theory [11], we have defined the effective vertices n (with an odd number n of external legs) as The series over r and the integral over t can be performed, and we finally get: Even though ( 16) is obtained for odd values of n, we will see below that the calculation of even Green's functions will also result in effective vertices Π (ǫ) n that are again given by (16) (with the exception of the special case n = 2).
From (14) we see that the Green's functions at O(ǫ) are written in terms of coordinateindependent (and then momentum-independent) effective vertices Π (ǫ) n of order ǫ, to which n external legs are attached.The effective vertex Π (ǫ) n is obtained once (at least part of) the interaction is integrated out, and as such it is a dressed vertex function.Later we will also introduce higher order effective vertices Π Even Green's functions.The O(ǫ) contributions to the even Green's functions come from L 1 0 , and coincide with the terms of the same order of the corresponding gφ 2 (φ 2 ) ǫ hermitian theory [11].Using the replica trick for the logarithm in L 1 0 we have As explained before, the functional integral is calculated by considering first integer values of N. For N < n 2 the path integral vanishes, as we cannot draw connected diagrams, while for N ≥ n 2 we have where C n (N) is the combinatorial factor coming from the contractions: As for N < n 2 the falling factorial (N) n 2 vanishes, Eq. (18) holds true even for N < n 2 , and then for all positive integers N. Starting from this observation, the required analytic extension of (18) to real values of N is obtained with the help of the identity (2N − 1)!! = 2 N −1 Γ(N + 1 2 ) Γ( 32 ) , and this finally allows to take the derivative with respect to N and the limit N → 1.
The O(ǫ) contribution to the even Green's functions G n is then where we defined the even n-legs O(ǫ) effective vertex Π Performing the derivative and the limit we finally obtain: where Comparing ( 23) with (16) we see that, as anticipated above, the two expressions for Π (ǫ) n with n even and odd coincide.
The introduction of these effective vertices provides a great advancement for several reasons.As for the corresponding ordinary theory [11], we will see that, at any order in ǫ, the Green's function can be written in terms of effective vertices and loop integrals.Moreover, we will show that the effective vertices are extremely useful for analysing the UV behaviour of the Green's functions at each order in ǫ.In the next subsection we start with the O(ǫ).

UV behaviour of the Green's functions at O(ǫ)
When studying a quantum field theory, it is of the greatest importance to analyse the UV behaviour of the Green's functions.We now perform such an analysis, and this is another novelty of the present section.To this end, we observe that the loop integral ∆(0) in (12), that appears in the Π (ǫ) n , diverges for d ≥ 2. From now on we are interested in studying these cases.
Starting with the one-point Green's function G 1 , that is nothing but the vacuum expectation value G 1 = 0|φ|0 = v, from (14) for n = 1 we see that: Needless to say, this divergence is trivially renormalized inserting in the lagrangian a linear counterterm δvφ.In this respect, we observe that, differently to what is claimed in [1], having got a non-vanishing negative imaginary result (actually negative imaginary infinite) for the unrenormalized G (ǫ) 1 is totally irrelevant in connection with the expectation that G 1 (as a result of the PT -symmetric nature of the theory) should be a negative imaginary number.The renormalization can produce any possible imaginary value, including G ren 1 = 0. Let us move now to the two-point Green's function G 2 .Going to momentum space, and adding to (20) the order ǫ 0 free propagator, up to O(ǫ) we have (remember that From ( 24) and ( 26) we see that the correction to the tree-level result for G 2 goes as: The term in the round brackets of ( 26) is easily recognized as the lowest order 1PI self-energy diagram within the expansion in ǫ.The resummation of the geometric series (in increasing powers of ǫ) can be then performed, thus obtaining for G 2 : Inserting ( 24) in (28), we see that the radiatively corrected mass m 2 R at this order is: ´, that diverges as log Λ for any d > 2, and as log(log Λ) for d = 2.This divergence is trivially cancelled once the mass counterterm contained in M 2 is tuned accordingly.Moreover it is worth to note that as compared to the algebraic divergence encountered at first order of the weak-coupling expansion, we here observe a much milder logarithmic divergence.Actually we have shown [11] that, when G 2 is calculated at the same order of approximation, this result for δm 2 is also present in the ordinary theory.This shows that such a mild UV behaviour is peculiar of the expansion in ǫ.
Going now to the G n with n > 2 (actually for n = 2 the equation below incorporates also the case in (25) for n = 1), from (23) we see that At this order then the theory turns out to be non-interacting, and this sounds as a surprising and disturbing result.This point clearly needs to be further investigated, and to this end we have to consider higher orders in the expansion.In the next section we will analyse systematically the O(ǫ 2 ).

Green's functions at O(ǫ 2 ) and their UV behaviour
The O(ǫ 2 ) contributions to the Green's functions are given by The L 2 term gives diagrams with one vertex, the other one diagrams with two vertices.We treat these two contributions separately, denoting them as respectively.

Diagrams with one Effective Vertex
From ( 5) we know that . Since L 2 0 and L 2 2 are even in φ, while L 2 1 is odd, they contribute only to the even and to the odd Green's functions respectively, so that it is useful to treat the two cases separately.
Odd Green's functions.Expliciting L 2 1 from (5), and using ( 6) and ( 7), we have Even Green's functions.The term L 2 2 is quadratic in φ, and then contributes only to G 2 , the calculation being trivial.Concerning the contribution of L 2 0 , from ( 5) and ( 6) we get: Performing in (32) and (33) the derivative with respect to N, the limit N → 1, the sum over r and the integral over t, for generic n (either even or odd) we get where we have defined the O(ǫ 2 ) effective vertices and K is given in (24), while H n stands for the n-th Harmonic number.In the expression for n = 2, we have also included the contribution from L 2 2 .
Analysis of the UV behaviour.Eq. ( 34) shows that the UV behaviour of the generic is entirely given by the UV behaviour of the Π 35) and (36), so that: Comparing ( 37) and ( 38) with the behaviour of the G n at O(ǫ) (Eqs.( 27) and ( 30)), we see that the leading behaviour of this one-vertex contribution is enhanced only by a power of log Λ (d > 2) or log (log Λ) (d = 2).We note that G 1 and G 2 get a divergent contribution, while G (ǫ 2 ,1) n vanishes when n ≥ 3.However, before drawing any conclusion, we have to consider the other O(ǫ 2 ) diagrams, namely the diagrams with two vertices.We do that in the next subsection.

Diagrams with two Effective Vertices
At O(ǫ 2 ), the diagrams with two vertices come from L 1 • L 1 terms.From (5) we see that L 1 = L 1 0 + L 1 2 , and then the product L 1 • L 1 splits in the sum of three contributions, the mixed term gets a factor two as there are two contributions that are equivalent under the exchange u 1 ⇐⇒u 2 ).The L 1 0 • L 1 1 term is odd in φ, and gives non-vanishing contributions only to the odd Green's functions, while the first and the third terms are even, and contribute only to the even Green's functions.Let us treat the two cases separately.Odd Green's functions.Taking from (5) the expression for L 1 0 • L 1 1 , and applying ( 6) and (7), we obtain: Let us evaluate the path integral in (39), starting by contracting the first s (≤ n) fields φ(x i ) with φ(u 1 ) and the remaining n − s fields with φ(u 2 ).The other diagrams are obtained by permutations of the external legs.We have to distinguish the case with an even number of internal lines, corresponding to an even number of external legs attached to the vertex u 1 (s = 2j), from the case with an odd number of external legs attached to u 1 (s = 2j + 1).These contributions will be denoted with G n, B respectively.Let us start with the A-type contribution.The connected diagrams that can be drawn from the path integral are of the kind (2l is the even number of internal lines) Note that 0 ≤ j ≤ n−1 2 .Moreover for generic integers r ≥ 0 and N > 0, connected diagrams are obtained only for l ≥ 1, and for: from which we also have that the number of internal lines is limited by Under these conditions, the path integral in (39), i.e. the sum of diagrams of the kind (40), gives (43) where C njl (N, r) is the combinatorial factor coming from the contractions: In (44) the terms C 2j+2l (N) and B n−2j+2l (r) are the combinatorial coefficients found at O(ǫ), when we were dealing with one-vertex diagrams with even and odd number of legs respectively.Concerning the conditions (41), it is important to make the following observations.Due to the presence of the falling factorial, the factor C 2j+2l (N), defined in (19), vanishes when the condition (41) 1 is not satisfied.We can then consider also values of N < l + j, and this allows to reach the value N = 1, that is necessary for the subsequent analytic extension.Moreover the factor B n−2j+2l (r), defined in (11), vanishes when 0 ≤ r ≤ n−1 2 − j + l, so that, despite the condition (41) 2 , we can take the sum over r in (39) starting from r = 0. Finally, thanks to these two latter observations, the sum over l in (43) can be extended up to infinity, as (44) vanishes when l > min.Then we can write We easily recognize in the two square brackets the O(ǫ) effective vertices Π k with even and odd number of legs, Eqs. ( 21) and (15) respectively, so that Eq. ( 46) is a compact and elegant form for the A-type contribution, written in terms of diagrams built with couples of effective vertices.Actually we will see that, at each order in ǫ, the G n are all written in terms of effective vertices (those of higher order in ǫ will be defined later) and loop integrals connecting them.This shows the great advantage induced by the effective vertices.
The series over l can be easily resummed, however we have to distinguish the case n = 1 from the others.Starting with the former we get: while for the others odd values of n: where the hypergeometric functions 2 F 1 and 3 F 2 are defined by (l) 2 and we denoted with z: Eqs. ( 47) and (48) show that the series over l give rise to hypergeometric functions that depend on the spacetime integration variables u 1 and u 2 .In this respect, we note that, as long as u 1 = u 2 , we have |z| < 1, and the convergence of the hypergeometric functions is guaranteed.However, when u 1 = u 2 , i.e. z = 1, convergence issues could arise.More precisely, for 2 F 1 the convergence is guaranteed if while for 3 F 2 the condition is Considering the arguments of the hypergeometric functions in (47) and (48) we see that both conditions (52) and (53) reduce to n < 5 . ( This means that, while G are certainly well-behaved, for n ≥ 5 we need to carefully analyse the singularities in G n, A induced by the limit (u 1 − u 2 ) → 0, and to investigate whether they are integrable or not.
Performing the explicit calculation for the cases d = 3 and d = 4, we have verified that starting from G (ǫ 2 ,2) 5, A more and more severe non-integrable singularities in the limit (u 1 − u 2 ) → 0 do actually appear.However, when the B-type contribution, i.e. diagrams with an odd number of internal lines, is also considered (see below), it turns out that the divergences introduced by these "A-diagrams" are exactly cancelled by opposite divergences due to the "B-diagrams".Similar cancellations also occur for the even Green's functions that we will consider later.For this reason, we do not need to consider any longer the singularities arising from the hypergeometric functions.
Let us move now to the B-type contribution.The connected diagrams that can be drawn are of the kind Following the same steps made for the A-type contribution, we obtain as before a final expression in terms of diagrams built with two O(ǫ) effective vertices Π k , one with even and one with odd number of legs: The series over l can be easily resummed.However, as before, we have to distinguish the case n = 1 from the others.Starting with the former we get: while for the other odd values of n and the final expressions for the odd Green's functions is obtained summing up G n, A and G (ǫ 2 ,2) n, B .Let us move now to the even Green's functions.Even Green's functions.Reading from (5) the expression for the terms L 1 0 • L 1 0 and L 1 1 • L 1 1 we obtain: We already encountered the first of the two terms in (59) when we calculated the G n at O(ǫ 2 ) for the ordinary scalar theory gφ 2 (φ 2 ) ǫ in [11].The contribution from this term is given by the sum of all the connected diagrams built with a couple of O(ǫ) effective vertices Π (ǫ) k with an even number of legs.Labelling with A 1 and B 1 the diagrams in which the number of external legs attached to each vertex is even and odd respectively, we have We now replace ( 22) and ( 23) for the effective vertices in (60) and (61), and sum the series over l.Starting with the two-point Green's function we get while for the even Green's functions with n ≥ 4 we have: We have to consider now the second of the two terms in (59).Using (7), we write this contribution to G (ǫ 2 ,2) n as: As before, we need first to calculate the connected component of the functional integral in (66) for each r 1 , r 2 ≥ 0. There are two classes of diagrams, depending on whether the number of external legs attached to each vertex is even (let us call them A 2 -type diagrams) or odd (B 2 -type diagrams), and we will treat these two cases separately.
Starting with the A 2 -type contribution, we have diagrams of the kind where for any couple r 1 , r 2 ≥ 0, the only possible diagrams are those with values of 0 ≤ j ≤ n 2 and l ≥ 0, satisfying the conditions which imply that the number of internal lines is limited by Under these conditions, the sum of the diagrams of the type (67) gives (70) where C njl (r 1 , r 2 ) is the combinatorial factor coming from the contractions: In ( 71) the terms B 2j+2l+1 (r 1 ) and B n−2j+2l+1 (r 2 ) are the combinatorial coefficients (11) found when we were dealing with O(ǫ) one-vertex diagrams with an odd number of legs.As for the odd Green's functions, this makes it possible to express the final result for the even G n in terms of diagrams with two effective-vertices Π (ǫ) k .Moreover, for the same reasons seen for the odd Green's functions, we can extend both the series over r 1 and r 2 starting from r 1 = r 2 = 0, and the sum over l up to infinity, regardless the conditions (68) and (69).We can then write (in this case there is no need to consider G 2 separately) where we easily recognize in the square brackets the O(ǫ) effective vertices (15) with an odd number of legs.Then Inserting in (73) the Π (ǫ) k as given in ( 16), and performing the sum over l, we get Moving to the B 2 contribution, we now see that the diagrams are of the kind Following steps similar to those made for the A 2 contributions, we obtain as before an expression in terms of diagrams built with two O(ǫ) effective vertices Π (ǫ) k with an odd number of legs: Inserting as before the Π k given in (16), and performing the sum over l: The O(ǫ 2 ) contribution to the even Green's functions from the two-vertex diagrams is then obtained summing up the four terms A 1 , A 2 , B 1 and B 2 .
Analysis of the UV behaviour.The analysis of the UV behaviour of the two-vertex contributions to the Green's functions is easily performed if we refer to (46), ( 56), ( 60 k (that is immediately read from ( 16) and ( 23)), and from the superficial degree of divergence of the loop integrals, we observe that in each of the series over l the first term is the dominant one, so that the leading behaviour of the different contributions are: where D Λ is the superficial degree of divergence of the loop integral with two propagators: Let us consider now the results (37), ( 38), ( 78), ( 79) and (80) for the UV behaviour of all the one-vertex and two-vertex O(ǫ 2 ) contributions to the Green's functions.We see that G 1 and G 2 diverge.As it was the case at O(ǫ), these divergences are trivially removed with the introduction of O(ǫ 2 ) linear and quadratic counterterms, together with an additional O(ǫ 2 ) wave function renormalization.This would be the starting point of a systematic renormalization of the theory at each order in ǫ.
On the other hand, we also see that the G n with n ≥ 3 vanish.Therefore, putting these results together with those obtained at O(ǫ), we conclude that up to O(ǫ 2 ) the theory is non-interacting.However, before drawing any conclusion on the original program of realizing a systematic renormalization of the theory within the framework of the logarithmic expansion, we have to consider the higher orders.

Higher order contributions to the G n
In this section we extend the analysis on the UV behaviour of the Green's functions to higher orders in ǫ.When moving to a generic order ǫ p , we encounter different contributions coming from diagrams having a number of vertices V ≤ p.
First of all we note that, referring to Eq. ( 5), the terms that contribute to G are given by products of L p i k i [φ(u i )], with i = 1, . . ., V .Each L p i k i brings a factor either of the type φ 2 log k φ 2 or of the type φ|φ| log k φ 2 , that have to be treated with ( 6) and (7).When performing the contractions, the combinatorial factors always split in coefficients of the kind B n and C n in (11) and (19), encountered when considered one-vertex diagrams (see (44) and (71) for the O(ǫ 2 ) two-vertex contributions).This gives rise to a factorization that finally leads to express the G (ǫ p ) n as a sum of all the possible diagrams built in terms of effective vertices.This general result is the great advancement brought by the introduction of the effective vertices.
As a consequence, the analysis of the UV behaviour of the Green's functions becomes an easy task even at higher orders in ǫ.We only need to derive the UV behaviour of the generic O(ǫ p ) effective vertex Π (ǫ p ) k , and the superficial degree of divergence of the loop integrals due to the propagators that connect these vertices.
To this end, let us first calculate the generic O(ǫ p ) effective vertices Π (ǫ p ) n by considering the one-vertex contribution to the G n at order ǫ p .From (5) we have Let us begin by considering even Green's functions, for which we only have even values of k, i.e. even powers of |φ| φ .In this case: By following similar steps to those employed in the previous sections, the path integral in (83) is immediately calculated.We get that diagrammatically is written as with the effective vertex Π Concerning the odd Green's functions, in (82) we only have odd values of k, i.e. odd powers of |φ| φ , so that in this case from the one-vertex diagram given by ( 84) and (88).In Sections 2 and 3 we have already seen that this is the case at order ǫ and ǫ 2 .Moreover, from (84) and (88) we see that the Green's functions G 1 and G 2 diverge at each order ǫ p .This is what we already found for the cases p = 1, 2. These divergences are trivially removed with the introduction of linear and quadratic counterterms at each order ǫ p .
Finally, (84) and ( 88) also tell us that the Green's functions G n with n ≥ 3 vanish for any dimension d ≥ 2. Therefore: At any finite order in ǫ, and for any d ≥ 2, the theory is non-interacting.This is a disturbing, and at first sight surprising, result.However it is not difficult to understand its origin.Actually, any finite order in ǫ gives nothing but an approximation to the interaction lagrangian that is truncated at a finite power of log(φ).Due to the mild behaviour of the logarithm (and of its powers), we can easily understand that this is a "too poor" truncation of the physical interaction, not sufficient to grasp enough of the quantum fluctuations, i.e. to guarantee the existence of non-trivial S-matrix elements.
It is worth to stress at this point that similar results were obtained by us for the corresponding gφ 2 (φ 2 ) ǫ ordinary (hermitian) theory [11].Performing the same expansion, we found that at each order in ǫ the theory gφ 2 (φ 2 ) ǫ is non-interacting.
This clearly shows that the fact that the theory gφ 2 (iφ) ǫ is non-interacting at each order in ǫ has nothing to do with its non-hermitian nature.It is rather an intrinsic weakness of the logarithmic expansion.As just stressed, the reason is that at any finite order in ǫ this expansion gives a "too poor" truncation of the interaction.
Moreover, the fact that at any order in ǫ the theory is non-interacting shows that the original program of a systematic renormalization of the theory at each order in ǫ [1] cannot be realized, actually looses its meaning.In particular, the renormalization of the field vacuum expectation value and of the mass, obtained through the introduction in the lagrangian of the linear and quadratic counterterms δv φ and 1 2 δm 2 φ 2 (considered above), has no physical significance.
Physically speaking, there is no way of giving a meaning to the theory if we truncate it to a finite order in ǫ.
We might still hope that a sensible definition could be given after resumming diagrams from each order in ǫ.In this respect, it is clear that the first thing to do is to resum the leading contributions to the Green's functions, that as we have previously shown are the one-vertex diagrams (84).This amounts at resumming the ladder of leading logarithms in (88).
Before ending this section we observe that, in order to perform the resummations, it is useful, although not necessary, to go back to Eq. (86) (and the analogous one for odd values of k) and close the sum over k.To this end, defining for n even i sin(Nπ) for n odd (93) and noting that lim we can write the generic n-legs effective vertex at O(ǫ p ) as Let us proceed now to the resummations.

Resummation of the one-vertex diagrams
In the previous sections we have seen that, at any given order ǫ p , the leading contributions to the Green's functions G n come from the one-vertex diagrams (85).
Indicating their resummation with G n : Eq. ( 96) shows that the resummation of the G (ǫ p ,1) n actually amounts to resumming the Π (ǫ p ) n , thus providing an x-independent (and then p-independent) approximation Γ n to the n-points vertex function Γ n .From Eq. (95) we see that: where the last step is possible as the function f n (N) is analytic on the positive real axis.From (93), we finally get for n even i sin( ǫπ 2 ) for n odd (98) Few comments are in order.First of all we note that the resummation of the leading logarithms has drastically changed the UV behaviour of the Green's functions.In particular the dependence on powers of logarithms (log p (∆(0)) or log p−1 (∆(0)), see Eq. ( 88)) has been traded by the algebraic dependence ∆(0) ǫ 2 .Interestingly, while at any finite order in ǫ, the UV behaviour of G n does not depend on ǫ, after resummation it does.
Restricting ourselves to integer values of ǫ, from Eq. (98) we see that, due to the presence of the functions cos( ǫπ 2 ) and sin( ǫπ 2 ), the G n for which n has not the same parity of ǫ vanish.In addition, as the denominator contains the function Γ ǫ+4−n 2 , only those G n with n < ǫ + 4 are non-vanishing.
However, the condition ǫ < 2 is necessary to restrict the field variable φ in the path integral that defines the Green's functions to real values, as required by the method followed in the present work.Therefore, among the possible integer values of ǫ, we can consider only ǫ = 1.As it corresponds to the largely considered PT -symmetric 1  2 igφ 3 theory (see the lagrangian (1)), this case is particularly relevant to study.L p−k .However, under the sum over k these two cases bring the same contribution, thus giving a factor 2 that cancels the factor 1  2 in (103).Therefore (as done in section 3.2) we distinguish two classes of diagrams, namely those where the two effective vertices are connected with an even number of internal lines, and those connected by and odd number of lines, indicating them as A-type and B-type diagrams respectively.We have: and that correspond to diagrams of the kind: the sum of all these two-vertex diagrams, and perform separately the resummation of A-and B-type diagrams.Let us begin with the former: where we introduced the notation (108) The double series in the curly brackets is nothing but the Cauchy product: where q = p − k, and we need that the two series in the r.h.s. are convergent, and at least one of them absolutely convergent.
We already summed in (98) the series in the round brackets.They are Taylor expansions of functions analytic in the complex half-plane Re(ǫ) > −3, due to the presence of Γ ǫ+3 2 in (98), and then their radius of convergence in ǫ is 3. Therefore the validity of (109) is guaranteed for ǫ < 3: in this case both series are absolutely convergent.
For our scopes Eq. ( 109) is crucial, as it converts the resummation of diagrams with two effective vertices in the product of series whose terms are single effective vertices.Then, thanks to (97), Eq. ( 109) results in the product of vertex functions Γ (1) n (given in (98)).From ( 107) and (109) we then have: Following similar steps, the contribution from B-type diagrams is: and summing (110) and (111), we finally get the odd G n .
Even Green's functions.When the number of external legs is even, there are two classes of diagrams: diagrams where the two effective vertices have an even number of legs (type 1), and diagrams where both the effective vertices have an odd number of legs (type 2).Moreover, in both cases there are two possibilities: each of the effective vertices can be connected either with an even (A-type diagrams) or with an odd (B-type) number of external legs.Then we have 4 different classes of diagrams: A 1 , B 1 , A 2 and B 2 (the same classes that we encountered in section 3.2).Summing these contributions from all orders in ǫ, and following similar steps to those made for the odd Green's functions, we find that all these contributions are easily expressed in terms of the resummed vertex functions Γ ô . (115) The sum of (112)-(115) finally gives the even G n .As in the previous section, we now specify to the interesting case of the 1 2 igφ 3 theory.We will see below that, as it was the case for the resummation of diagrams with one effective vertex, again we get simple and trivial results of the weak-coupling expansion.
Going to momentum space, and considering as usual amputated Green's functions, from each of the contributions in (110)-(115) we get (the vertices with three and two legs of the weak-coupling expansion were already identified in the previous section): itself, we simply get the trivial O(g) results of the weak-coupling expansion.
Having considered the resummation of diagrams with a single effective vertex, we next moved to resum diagrams with two effective vertices, but again we got trivial results of the weak-coupling expansion.The outcome of these resummations has not really improved the status of the logarithmic expansion, as far as it concerns the possibility of using it to get non-trivial/non-perturbative results, as it was originally expected (hoped) [1,7,8,10].
Naturally, there might still be the possibility that other resummations could provide less trivial results.In any case, what we have definitely established with the present work is that the original idea to use the expansion in ǫ to implement a systematic renormalization of the theory cannot be implemented, actually it has no sense.