Towards perturbative renormalization of $\phi^2(i\phi)^\varepsilon$ quantum field theory

In a previous paper it was shown how to calculate the ground-state energy density $E$ and the $p$-point Green's functions $G_p(x_1,x_2,...,x_p)$ for the $PT$-symmetric quantum field theory defined by the Hamiltonian density $H=\frac{1}{2}(\nabla\phi)^2+\frac{1}{2}\phi^2(i\phi)^\varepsilon$ in $D$-dimensional Euclidean spacetime, where $\phi$ is a pseudoscalar field. In this earlier paper $E$ and $G_p(x_1,x_2,...,x_p)$ were expressed as perturbation series in powers of $\varepsilon$ and were calculated to first order in $\varepsilon$. (The parameter $\varepsilon$ is a measure of the nonlinearity of the interaction rather than a coupling constant.) This paper extends these perturbative calculations to the Euclidean Lagrangian $L= \frac{1}{2}(\nabla\phi)^2+\frac{1}{2}\mu^2\phi^2+\frac{1}{2} g\mu_0^2\phi^2\big(i\mu_0^{1-D/2}\phi\big)^\varepsilon-iv\phi$, which now includes renormalization counterterms that are linear and quadratic in the field $\phi$. The parameter $g$ is a dimensionless coupling strength and $\mu_0$ is a scaling factor having dimensions of mass. Expressions are given for the one-, two, and three-point Green's functions, and the renormalized mass, to higher-order in powers of $\varepsilon$ in $D$ dimensions ($0\leq D\leq2$). Renormalization is performed perturbatively to second order in $\varepsilon$ and the structure of the Green's functions is analyzed in the limit $D\to 2$. A sum of the most divergent terms is performed to {\it all} orders in $\varepsilon$. Like the Cheng-Wu summation of leading logarithms in electrodynamics, it is found here that leading logarithmic divergences combine to become mildly algebraic in form. Future work that must be done to complete the perturbative renormalization procedure is discussed.

In a previous paper it was shown how to calculate the ground-state energy density E and the p-point Green's functions Gp(x1, x2, ..., xp) for the PT -symmetric quantum field theory defined by the Hamiltonian density H = 1 2 (∇φ) 2 + 1 2 φ 2 (iφ) ε in D-dimensional Euclidean spacetime, where φ is a pseudoscalar field. In this earlier paper E and Gp(x1, x2, ..., xp) were expressed as perturbation series in powers of ε and were calculated to first order in ε. (The parameter ε is a measure of the nonlinearity of the interaction rather than a coupling constant.) This paper extends these perturbative calculations to the Euclidean Lagrangian L = 1 2 (∇φ) 2 + 1 2 µ 2 φ 2 + 1 2 gµ 2 0 φ 2 iµ which now includes renormalization counterterms that are linear and quadratic in the field φ. The parameter g is a dimensionless coupling strength and µ0 is a scaling factor having dimensions of mass. Expressions are given for the one-, two, and three-point Green's functions, and the renormalized mass, to higher-order in powers of ε in D dimensions (0 ≤ D ≤ 2). Renormalization is performed perturbatively to second order in ε and the structure of the Green's functions is analyzed in the limit D → 2. A sum of the most divergent terms is performed to all orders in ε. Like the Cheng-Wu summation of leading logarithms in electrodynamics, it is found here that leading logarithmic divergences combine to become mildly algebraic in form. Future work that must be done to complete the perturbative renormalization procedure is discussed.

I. INTRODUCTION
Since the publication of the first paper on PT symmetry in 1998 [1], in which the PT -symmetric quantummechanical Hamiltonian was introduced, this research area has become highly active. This model has been studied in detail [2], and much theoretical research has been done on the mathematical structure of non-Hermitian quantum systems [3]. Beautiful experiments have been performed in diverse areas of physics including optics, photonics, lasers, mechanical and electrical analogs, graphene, topological insulators, superconducting wires, atomic diffusion, NMR, fluid dynamics, metamaterials, optomechanical systems, and wireless power transfer [4][5][6][7][8][9][10][11][12][13][14][15][16][17]. This paper considers the generalization of (1) to quantum field theory in D-dimensional Euclidean space. In an earlier paper [18] we examined the corresponding fieldtheoretic Lagrangian density where φ is a (dimensionless) pseudoscalar field. Note that (2) is manifestly PT -symmetric because φ → −φ under space reflection P and i → −i under time reversal T . Just as (1) is a testbed of PT -symmetric quantum mechanics, (2) is a natural model for the study Ddimensional PT -symmetric bosonic field theories. The Hamiltonian (1) launched the field of PTsymmetric quantum theory because it has the surprising feature that if ε ≥ 0, its eigenvalues are all discrete, real, and positive even though it is not Dirac-Hermitian [19,20]. (A Dirac-Hermitian Hamiltonian obeys the symmetry constraint H = H † , where † indicates combined complex conjugation and matrix transposition.) Moreover, the quantum theory defined by (1) is unitary (probability conserving) with respect to the adjoint CPT , where C is a linear operator satisfying the three simultaneous operator equations [21] C 2 = 1, [C, PT ] = 0, [C, H] = 0. ( Thus, while the constraint that a Hamiltonian be Dirac-Hermitian is sufficient to define a consistent quantum theory, it is not necessary. In short, PT -symmetric quantum theory is not in conflict with the axioms of conventional Hermitian quantum theory; rather, it is a complex generalization of Hermitian quantum theory. Almost all of the theoretical research on PT symmetry has focused on PT -symmetric quantum mechanics and experimental studies of PT -symmetric classical systems, but very few papers have focused on PT -symmetric quantum field theory. We briefly review some of the earlier work on PT -symmetric quantum field theory: 1. Early studies of PT -symmetric quantum field theory considered the case ε = 1 in (2) [22,23]. This iφ 3 quantum field theory had emerged in studies arXiv:2103.07577v1 [hep-th] 13 Mar 2021 of Reggeon field theory [24] and the Lee-Yang edge singularity [25]. Conventional diagrammatic perturbation theory can be used to treat this cubic interaction: One introduces a coupling constant g in the interaction term igφ 3 and expands the physical quantities in powers of g. However, one cannot use conventional perturbation theory for other values of ε > 0, integer or noninteger.
2. PT -symmetric electrodynamics also has a cubic interaction [26]. The Johnson-Baker-Willey (JBW) program for constructing a finite massless electrodynamics fails because the zero of the beta function yields at best a negative and perhaps a complex value of α because conventional Hermitian quantum electrodynamics (QED) is not asymptotically free. However, the JBW procedure works for the PT -symmetric version of QED because this theory is asymptotically free and one obtains a reasonable positive numerical value for α.
3. Renormalizing a Hermitian quantum field theory often causes the Hamiltonian of the theory to become non-Hermitian. This problem was observed in the Lee model [27], which is again a theory with a cubic interaction. Pauli and Källén showed that upon renormalization, ghost states (states of negative norm) arise, and appear to violate unitarity in scattering processes. This problem remained unresolved until 2005, when it was shown that if one uses the appropriate PT -symmetric inner product for the renormalized Lee-model Hamiltonian, there are no ghost states and the unitarity of the theory becomes manifest [28].
4. Renormalizing the Hamiltonian for the Standard Model of particle physics induces what appears to be instability in the vacuum state. This is due to the contribution of the top-quark loop integral [29]. Once again, the renormalized Hamiltonian appears to be non-Hermitian. However, by using PT -symmetric techniques it was shown by using a simple model-field-theory argument that the vacuum state and the next few higher-energy states are actually stable (have real energy) [30].
5. Introducing higher-order derivatives in a quantum field theory in order to make Feynman integrals converge also makes the Hamiltonian appear to be non-Hermitian. (Higher-order derivatives induce Pauli-Villars ghosts.) However, PT -symmetric techniques resolve this problem. The simplest fieldtheory model that exhibits this problem is the Pais-Uhlenbeck model and PT -symmetric techniques demonstrate that this theory has no ghosts [31].
6. The double-scaling limit in quantum field theory [a correlated limit in which the number N of species in an O(N )-symmetric field theory approaches infinity as the coupling constant approaches a critical value] appears to lead to an unstable field theory. For a quartic scalar field theory the value of g crit is negative and one might think that the resulting −φ 4 theory is unstable. However, the techniques of PT -symmetric quantum theory demonstrate that such a theory is actually stable and has a real positive spectrum [32,33]. It was observed by K. Symanzik that a −φ 4 theory is asymptotically free even though it does not have a local gauge symmetry but he called this theory "precarious" because it appears to be unstable.
7. Time-like Liouville field theories appear to be unstable and non-Hermitian but the techniques of PT -symmetric quantum field theory can be used to argue that the Hamiltonians of such theories have real spectra and induce unitary time evolution [34].
8. Studies of the PT -symmetric Dirac equation suggest that one may have species oscillations and still have massless neutrinos [35,36]. Interestingly, recent measurements in Germany have halved the upper bound on the mass of the electron neutrino [37].
In the studies above some striking results were obtained but many of these field-theory papers considered only simple zero-dimensional toy models and onedimensional quantum-mechanical analogs that suggest the possible behaviors of PT -symmetric field theories. Thus, there is strong motivation for developing general methods for solving quantum field theories defined by non-Hermitian PT -symmetric Hamiltonians.
Until recently, PT -symmetric quantum field theory has remained beyond the reach of comprehensive analytical study because of three technical problems that had to be overcome when trying to solve the PT -symmetric quantum field theory in (2): 1. Feynman perturbation theory works for the cubic case ε = 1 in (2) but for other integer values of ε the Feynman diagrams must be supplemented by nonperturbative contributions, which are nontrivial. Moreover, when ε is noninteger, there are no Feynman rules at all so one cannot perform a conventional coupling-constant expansion. 3. In PT -symmetric quantum theory one must use the C operator, obtained by solving (3), in order to obtain matrix elements. It is difficult to calculate C, so the prospect of calculating Green's functions appears to be rather dim.

The functional integral for the partition function
In Ref. [18] it is shown how to overcome these three problems by extending the methods developed in Refs. [39][40][41] for Hermitian quantum field theories to non-Hermitian PT -symmetric quantum field theories. In brief, instead of using a coupling-constant expansion, a perturbation expansion in powers of the parameter ε, which is a measure of the nonlinearity of the theory, is performed and summation methods are then used to evaluate the ε series. For small ε the functional integral converges on the real axis in complex-field space, and thus multidimensional Stokes sectors are not required for convergence. This solves problems 1 and 2 above. Expanding in powers of ε introduces complex logarithms in the functional integrand and PT symmetry is then enforced by defining the complex logarithm properly: The logarithms are now real, and the techniques for handling these logarithms to any order in powers of ε are based on methods that were introduced in Refs. [39][40][41]. The apparent problem with the C operator is actually not a problem if we are calculating Green's functions. This is because in a theory with an unbroken PT symmetry the vacuum state is an eigenvalue of C with eigenvalue 1: C|0 = |0 . Since the Green's functions are vacuum expectation values, we may ignore problem 3 entirely. This observation was first made in Ref. [42].
Why does L in (2) define an interesting theory? Let us first look at the cubic case. For a conventional gφ 3 theory the ground-state energy density E is a sum of 3vertex vacuum-bubble Feynman diagrams, and the Feynman perturbation expansion has the form This series diverges and the coefficients A n all have the same sign. Thus, if we perform Borel summation, we find a cut in the Borel plane, which implies that E is complex. Thus, the vacuum state is unstable. However, to obtain the PT -symmetric cubic theory we replace g by ig. Now, the perturbation expansion alternates in sign and the Borel sum of the series is real, the vacuum is stable, and the spectrum of the theory is bounded below. The case of the quartic PT -symmetric theory is more elaborate. The ground-state energy density for a conventional Hermitian quartic gφ 4 quantum field theory has a Feynman perturbation expansion of the form Again, this series is divergent, and since the perturbation coefficients B n all have the same sign, the series is alternating and Borel summable. The Borel sum of the perturbation series yields a real value for the vacuum energy density. This implies that the conventional quartic theory has a stable ground state, as one would expect. It may seem that the PT -symmetric quartic theory obtained by replacing g with −g is problematic because the perturbation series no longer alternates in sign: If we Borel-sum the series, we find a cut in the Borel plane, which suggests that E is complex and that the vacuum state is unstable, as one might intuitively expect with an upside-down potential. This conclusion is false! If one examines the functional integral for the partition function of the theory, one sees that the perturbative contribution to E (the Feynman diagrams) must be supplemented by imaginary nonperturbative contributions arising from two saddle points in complex function space. These additional pure-imaginary contributions exactly cancel the discontinuity in the Borel plane [43]. Consequently, the vacuum state for a PT -symmetric −gφ 4 theory is stable. We emphasize that Feynman diagrams alone are not sufficient to calculate the Green's functions of a PT -symmetric quantum field theory.
The research objectives in this paper are to extend the work in Ref. [18] and to study in depth the problem of renormalization. In Ref. [18] the Lagrangian (2) was examined to first order in ε. Treating ε as a small perturbation parameter, L was expanded in a series, which to first order in ε is Identifying the free Lagrangian as we developed techniques to evaluate the shift in the ground-state energy density ∆E and the Green's functions G p to first order in ε. We found that We then found that the two-point connected Green's function in momentum space to first order in ε is Thus, the renormalized mass to order ε is In addition, the higher-order connected Green's functions were also calculated to first order in ε: where the free propagator ∆ λ (x) associated with L 0 = 1 2 (∇φ) 2 + 1 2 λ 2 φ 2 obeys the general D-dimensional Euclidean Klein-Gordon equation The solution to (11), has the property that The corresponding selfloop is then Selfloop factors with λ = 1, which is associated with L 0 in (6), enter into (7), (8), and (10) and lead to the factors of Γ 1 − 1 2 D that occur in these expressions. This calculation is verified in D = 0 and D = 1 in Ref. [18], where exact calculations are possible, and provides confidence in these perturbative results.
To proceed with the renormalization program we must overcome two problems. First, we must calculate the Green's functions to higher order in ε. Second, we must show how to renormalize these Green's functions perturbatively for D ≥ 2. This paper is focused on renormalization in two dimensions. Specifically, we begin with the formulas for the one-point Green's function G 1 in (8) and for the square of the renormalized mass M 2 R in (9). These quantities are finite for 0 ≤ D < 2. However, as D approaches 2 from below (D → 2 − ) they diverge because Γ(z) has a pole at z = 0: and the formulas for G 1 (ε) and M 2 R simplify to where . From (10) we see that the Green's functions G p (p > 2) vanish as δ → 0, so the theory becomes noninteracting to order ε at D = 2.
The question is whether perturbative renormalization can be accomplished in the context of an expansion in powers of the parameter ε. Ordinarily, for interacting bosonic field-theories, renormalization is performed in the context of a coupling-constant expansion. Here, we perform expansions in powers of ε, which is a measure of the nonlinearity of the selfinteraction. Series expansions of this type were introduced many years ago [39] (prior to the study of PT symmetry), but the problem of renormalization has not been addressed until now.
In our renormalization program, we use the information gained from (15) and (16). The one-point Green's function G 1 , which is not directly measurable, becomes infinite as δ → 0. We can remove this divergence by introducing in the Lagrangian a linear counterterm ivφ, where v has dimensions of (mass) 1+D/2 and v = v 1 ε + v 2 ε 2 + v 3 ε 3 + · · · . Such a term is consistent with PT symmetry if v is real: Under PT reflection both the pseudoscalar field φ and i change sign. In addition, the divergence in (16) suggests that we should introduce an (infinite) mass counterterm µ (the unrenormalized mass) into the Lagrangian. Perturbative mass renormalization then consists of expressing the renormalized mass M R in terms of these Lagrangian parameters and absorbing into the parameter µ the divergence that arises as δ → 0.
Thus, we consider the Lagrangian density that now contains a dimensional field φ, the dimensional parameters µ, v, and µ 0 (a fixed parameter having dimensions of mass), and the dimensionless unrenormalized coupling g. Our objective is to calculate Green's functions for this quantum field theory as series in powers of the parameter ε and then to carry out perturbative renormalization for the two-dimensional case. This is a nontrivial extension of the earlier work in which the Green's functions were calculated to leading order in powers of ε [18] for the dimensionless Lagrangian density (2). We thus generalize (2)-(16) to first order in ε. The formula for G 1 in (8) is modified to read and the renormalized mass (9) becomes where in both expressions we introduce the dimensionless quantity m 2 = g + µ 2 /µ 2 0 and we display the new parameters explicitly in terms of ∆ 1 (0), whose behavior as δ → 0 is given by (14).
The formulas for G 1 (ε) and M 2 R simplify to , we remove the divergence in G 1 as δ → 0. Next, we see from (21) that the renormalized mass M R is logarithmically divergent as D → 2 − . We absorb this divergence into the mass counterterm µ by setting where B is a constant having dimensions of (mass) 2 .
(Note that the µ 2 counterterm is large and negative.) The renormalized mass is now finite, M 2 R = A + B, and the constant B is in principle determined from the experimental value of the renormalized mass M R .
In this paper we study the expansions of the Green's functions to higher-order in ε and examine perturbative renormalization in the limit δ → 0. In Sec. II we use techniques developed in [18] to calculate the connected Green's functions to order ε 2 . We examine the effects of the infinite linear and quadratic counterterms ivφ and µ 2 φ 2 on the Green's functions to second order in ε. In Sec. III we perform a multiple-scale analysis in which we obtain the leading contribution to each order in ε and sum these terms to all orders in ε. We compare the results as δ → 0 with those obtained via perturbative renormalization. Conclusions are given in Sec. IV.

II. GREEN'S FUNCTIONS TO ORDER ε 2
The p-point Green's function G p is defined as where the Lagrangian (17) enters both in the exponential function in the numerator and in the partition function in the denominator. Expanding the interaction term in powers of ε, we get where The Green's function G p is not connected and to solve the renormalization problem we require the connected ppoint Green's functions, which are constructed from cumulants. The procedure to order ε is explained in detail in Ref. [18] but to second order the cumulants become quite complicated. Of course, the appropriate connected graphs are easy to identify if the expression for G p contains polynomials and not logarithms of the field φ, and in this case standard techniques can then be used to evaluate the graphs. Our strategy here is to recast (24) into a form containing only products of the field φ at different space-time points. [The expression (24) does not include higher-order terms in ε that arise from expanding the denominator Z in powers of ε, as these lead to disconnected diagrams.] From here on, in our Green's function calculations we always discard disconnected contributions to the Green's functions, and we use the notation G p to represent the connected Green's functions.
Let us consider terms in G p (ε; y 1 , ...y p ) to order ε 2 : where Thus, in the expansion of the connected Green's functions in powers of ε, G p (ε; y 1 , ..., y p ) = ∞ n=0 ε n G p,n (y 1 , ..., y p ), we identify and so on, where we have suppressed the arguments y 1 , ..., y p on the left sides.
A. One-point Green's function to second order The one-point Green's function can be calculated by adopting the techniques developed in Ref. [18]. Terms odd under φ → −φ integrate to zero, so G 1,0 = 0, and G 1,1 , the term proportional to ε, is a constant independent of y 1 . The solution, generalized from Ref. [18], is which is given in (18). In Ref. [18] it is explained how to treat the term I 1 in (28). This term contains a complex logarithm in (27). The complex logarithm is converted to a real logarithm via (4) and the real logarithm is then treated by applying the replica trick [48] log where the mass constant µ 0 keeps the equation dimensionally consistent.
A second insight in [18] concerns terms containing the structure i|φ|/φ in (4). Here, we use the integral identity |φ|/φ = (2/π) ∞ 0 (dt/t) sin(tφ) and then expand sin(tφ) as a Taylor series in powers of φ: [Note that the integration variable t has dimensions of (mass) 1−D/2 so this equation is dimensionally consistent.] Thus, the leading term in the ε expansion of G 1 (ε) is proportional to ε and was determined to be (18). The coefficient of ε 2 in the expansion of G 1 (ε) in (28) is a sum of two functional integrals G 1,2 = A 1 + A 2 : To evaluate A 1 we insert (4) into (32) to obtain real logarithms and then insert the factor of φ 2 (x). Discarding terms odd in φ, we get Next, we use (30) and (31) to replace |φ(x)| and the logarithm by a sum over products of fields: We use graphical techniques to do the functional integral: A 1 contains products of the fields φ(y 1 )φ 2ω+2N +3 (x) and represents a free propagator connecting y 1 to x in (2ω + 2N + 3) ways, multiplied by products of selfloops from x to x; there are ω+N +1 selfloops. The functional integral yields the value ∆(y 1 − x)∆ ω+N +1 (0)(2ω + 2N + 3)!! [For brevity we suppress the mass subscript λ = µ 0 m in the free propagator ∆ defined in (11).] We can now perform the D-dimensional integral over x and use (12): (Note that the translation invariance of G 1 (ε) holds to second order in ε.) Combining these results, we get Taking the derivative with respect to N and the limit N → 0, this simplifies to where we use the duplication formula for the digamma function ψ(2z) = 1 2 ψ(z) + 1 2 ψ z + 1 2 + log 2. To evaluate the sum and integral we use [18] To calculate the second term in (33) we insert the integral representation [49] perform the sum over ω and the integral over t, and then note that the z integral gives the factor ψ(2). Thus, Combining (34) and (35), we obtain A 1 from (33): Next, we evaluate A 2 in (32). Expanding I 2 1 produces fields φ at two points, say x 1 and x 2 , which are integrated over. As before, we replace each occurrence of an imaginary logarithm by using (4) and retain terms that are even in φ. (The functional integral vanishes for terms that are odd in φ.) Only one term survives: Using (31) to replace φ(x 1 )|φ(x 1 )| and the replica trick (30) to replace the logarithm, we express A 2 as: Here we must evaluate the functional integral that connects the field at the external point y 1 to one of the (odd number of) fields at the internal points x 1 ; the remaining (even number of) points must be connected to the even number of points at x 2 . To this we must add the result of connecting the external point y 1 to the (even number of) points at x 2 leaving an odd number of connections from x 2 to x 1 . This includes all connected diagrams illustrated schematically in Figs. 1(a) and 1(b). The path integral corresponding to Fig. 1(a), for the field combinations φ(y 1 )φ 2ω+3 (x 1 )φ 2N +2 (x 2 ) allows y 1 to connect to one of the 2ω + 3 replicas of φ at x 1 . For the remaining points to be connected, 2l lines can connect the remaining 2ω + 2 points at x 1 to the 2N + 2 points at x 2 ; the remainder of points (2N + 2 − 2l) + (2ω + 2 − 2l) are used to form N + ω + 2 − 2l closed loops. Here, l ≥ 1 for the graph to be connected. We assign to Fig. 1(a) the combinatoric factor For Fig. 1(b) there are 2N +2 ways to connect y 1 to x 2 . To create connected graphs an odd number, say 2l + 1, of lines must join the remaining 2N + 1 points at x 1 to the 2ω +3 points at x 2 , where the minimum value of l is zero. Closed loops can exist on 2N + 1 − (2l + 1) = 2N − 2l points on x 2 and 2ω + 3 − (2l + 1) points on x 1 , forming a total of N + ω − 2l + 1 loops. The combinatoric factor assigned to this graph is It is best to examine the contributions from Fig. 1(a) and Fig. 1(b) separately. We use the combinatoric factor C a associated with Fig. 1(a) to evaluate its contribution to (37) and perform one spatial integral: Exchanging the sums on l and ω with ω ≥ l − 1, we determine the sum over ω: where we have shifted the summation variable to p = ω + 1 − l, which runs from 0 to ∞, and we have used the sums p z p /p! = e z , p pz p /p! = ze z , and p p 2 z p /p! = (z 2 + z)e z .
Setting z = 1 2 ∆(0)t 2 we evaluate the integral over t: Thus, (38) becomes To evaluate the sum over l we simplify the factorials and Gamma functions by using the duplication formula Γ(2z) = Γ(z)Γ z + 1 2 2 2z−1 / √ π and Euler's reflection formula Γ(z)Γ(1 − z) = π/ sin(πz). Then (40) reduces to where 2 F 1 a, b; c; x 2 is a Gaussian hypergeometric function [49]. When N = 0 this function becomes simply 2 F 1 − 1 2 , −1; 1 2 ; x 2 = 1 + x 2 and lim N →0 Next, evaluate the contribution from Fig. 1(b). Multiplying the combinatorial factor C b with the associated propagators in the evaluation of the functional integral of (37) and performing one spatial integration, we get Exchanging orders of summation over l and ω and using (39) to sum over ω and integrate over t, we obtain Again, on applying the duplication and reflection formulas for the Gamma function, we find that this expression also simplifies to the compact form When N = 0, the hypergeometric function simplifies, 2 F 1 − 1 2 , 0; 3 2 ; x 2 = 1, and its derivative at Fig. 1b] is constructed by adding (41) and (42), yielding the formal result The integrals containing linear and quadratic powers of ∆(x) are readily evaluated, the linear one following from (12), and the quadratic one from the solution to (10), using . The connected part of G 1 to order ε 2 is A 1 + A 2 , so we add the above result to (36) to get The first term is proportional to the dimensionless coupling strength gm −2 while the second term is proportional to its square. In the second term the expression in curly brackets is a dimensionless number. We now examine the limit D → 2 (that is, δ → 0). From the solution to (11) and (13) we see that the combination ∆(x)/∆(0) ∼ δK 0 (µ 0 m|x|) to first order in δ as δ → 0. Thus, to lowest order in δ the last term containing the integral simplifies to . Taking the same limit in (29), the coefficient of G 1 to first order in ε yields G 1,1 → − 1 2 igm −2 δ −1/2 . Putting these two last results together, we find that the divergence structure of G 1 in the ε expansion has the form where c 1 = 1 2 gm −2 , c 2 = 1 4 gm −2 ψ(2)−log π −gm −2 1+ ψ 3 2 − log π , and c 3 = − 1 4 gm −2 (1 − gm −2 ) are constants. So, the algebraic divergence δ −1/2 occurs at each order in the ε expansion. Our key result is that the second-order term in the ε expansion introduces log δ but does not alter the algebraic structure of the divergence.
Inclusion of the counterterm −ivφ. As seen from the calculations above, G 1 is negative imaginary and diverges in two dimensions as δ → 0. Since G 1 is not a physically measurable quantity, it can be removed by introducing an appropriate counterterm −ivφ into the Lagrangian, To examine the effect of including such a term, we denote the one-point connected Green's function associated with the full Lagrangian that includes −ivφ as G 1 (v). Its relation to the Green's functions evaluated without v is determined as follows. [We keep the notation of this section, without explicitly writing v = 0; that is, G 1 (v = 0) = G 1 and similarly for the higherorder Green functions and all expansion coefficients.] We include the contributions to the path integral of to second order in ε. This multiplies the expansion in ε of G p in (26). Identifying terms proportional to ε and ε 2 and using the definitions in (28), we find that for any p where we have suppressed the arguments y 1 , ..., y p of G p,n . Thus, the coefficients of the ε expansion G p (v) = n ε n G p,n (v) are related to the coefficients of the expansion of G p through the above expression. In principle, if we calculate the coefficients in the ε expansion of the Green's function without v, we can easily determine the effects of including it.
Let us evaluate G 1 (v). From (43) we get since G p,0 = 0 unless p = 2. Thus, to evaluate G 1 (v) to second order in ε, we must know the expansion coefficients of the two-point Green's function, calculated to first-order in ε without v. But these are known; by definition, G 2,0 (y, x) = ∆(y − x) while ∆(0)/2 is a generalization of the result obtained in [18] that includes the mass parameter µ 0 m of the Lagrangian L 0 used here [see (25)]. In the limit δ → 0, G 1,1 → −igm −2 /(2 √ δ) from (29), so in this limit Setting G 1 (v) = 0, we obtain the first-order result in (20), which fixes v 1 → gµ 2 0 /(2 √ δ). If we insert this into (45), then v 2 in turn is fixed to eliminate the O ε 2 term. It too has the same divergence structure v 2 → δ −1/2 . In summary, from our O ε 2 calculation of G 1 , we find that the divergence structure obtained when D → 2 has the same algebraic form that was determined from the O(ε) term; namely, δ −1/2 . It is accompanied by a logarithmic divergence in δ. This is a structure that persists for higher-order Green's functions. We will see that as D → 2, the algebraic structure of the lowest-order Green's functions is accompanied by logarithmic divergences as one goes to higher orders in the ε expansion.
Before applying these, expressions containing I n will have the occurrence of log n iµ (27) written in terms of real logarithms via (4) and expanded in a binomial series. As illustrated in detail in the previous section, the aim is to reduce the expressions that arise to products of functional integrals containing powers of φ multiplied by appropriate factors. From (28) the lowest-order contribution to G 2 (y 1 , y 2 ) is the free Green's function G 2,0 (y 1 , y 2 ) = ∆(y 1 − y 2 ). Performing the operations delineated in the last section, the coefficient of ε becomes where the connected graph associated with the functional integral connects the two external points to one internal one (see Fig. 2). In this case the functional integral takes the value [(2N + 2)!/(2 N N !)][∆(0)] N ∆(y 1 − x)∆(x − y 2 ), leading to the final result given in (44).
In the ε expansion of G 2 (y 1 , y 2 ), the coefficient propory 2 x y 1 Figure 2: Graphical representation of connected terms contributing to the functional integrals in (46) and (47).
tional to ε 2 , has the form G 2,2 = A 2;1 + A 2;2 , where For A 2;1 the even terms in the expansion of the imaginary logarithm give rise to the connected terms joining y 1 and y 2 , and thus the functional integrals that arise correspond to those depicted in Fig. 2, where x is an internal vertex that is integrated over. A detailed calculation gives where For A 2;2 , the square of the integral in (48) leads to two internal points x 1 and x 2 to which y 1 and y 2 can be connected. The possible connected diagrams are shown in Fig. 3 (48). In 3(a), y 1 (y 2 ) connects to the point x 1 (x 2 ), with the remaining number of points connecting x 1 to x 2 . In 3(b) the connection of y 1 (y 2 ) is to the points at x 2 (x 1 ), with the remaining points connecting to x 1 to x 2 . In 3(c) and 3(d) both y 1 and y 2 are connected to the same point, either x 1 or x 2 , with these being connected. x 1 and x 2 are integrated over.

(a)-3(d). A detailed calculation yields
where is a dimensionless constant and has dimension (mass) D . Collecting all terms for G 2,0 , G 2,1 , and G 2,2 , we obtain G 2 (y 1 , y 2 ) = G 2 (y 1 − y 2 ): where we have abbreviated Since Q 1 (y 1 − y 2 ) and Q 2 (y 1 − y 2 ) are convolutions, their Fourier transforms give products of the Fourier transforms of their components. Thus, the Fourier transform of G 2 (y 1 − y 2 ) takes the form The renormalized mass M 2 which reduces to (19) to first order in ε. On substituting m 2 = g + µ 2 /µ 2 0 , we get Divergence structure as D → 2. The divergence structure of G 2 (y 1 − y 2 ) when D → 2 (δ → 0) can be determined from (49). In this limit, ∆(x) → f 1 (x) + δf 2 (x) while K 1 diverges as − 1 2 log δ + 3 2 − γ 2 − 1 2 log(4π) and K 2 andf (0) introduce divergences of order (log δ) 2 . Putting this together, we find that where constant prefactors are suppressed and c n are constants. Evidently, higher-order terms in the ε expansion for G 2 (y 1 − y 2 ) display the same algebraic divergence as the lowest-order term, with higher-order corrections being logarithmic. This was previously observed for G 1 and we believe it to be generally true.
In summary, if we expand G 2 (y 1 − y 2 ) first in terms of ε, treat δ as finite, perform a Fourier transform, and invert it to identify a renormalized mass, we see that each coefficient in the ε expansion for M 2 R also diverges. The term in M 2 R proportional to ε diverges as log δ, while the structure of the terms proportional to ε 2 introduces divergences of up to log 2 δ. Thus, in such a perturbative calculation, the mass counterterm µ must absorb divergences that arise at each order of ε.

C. Three-point Green's function in second order
The connected three-point Green's function can also be calculated up to second order, using the techniques of the last sections. As this is tedious, we only give final results. The first-order coefficient in the ε expansion is G 3,1 (y 1 , y 2 , y 3 ) = −igµ 2 0 π 2∆(0) R(y 1 , y 2 , y 3 ), where R 1 (y 1 , y 2 , y 3 ) = d D x∆(x − y 1 )∆(x − y 2 )∆(x − y 3 ).
The ε 2 coefficient of G 3 in the ε expansion is where In this paper we apply MSPT techniques to the PTsymmetric Lagrangian (17) and we show what happens if we sum to orders in ε the most divergent perturbative contributions to the Green's functions. These contributions are logarithmic; this is not surprising because we are expanding in powers of ε, which is a parameter in the exponent.
We can identify the leading terms in the ε expansion of the Green's functions by restricting our attention to evaluating terms for the p-point connected Green's function that connect to only one internal point x. Then, the expansion for the Green's function includes only those terms arising from with I n = − 1 2 d D x φ 2 (x) log n iµ 1−D/2 0 φ(x)] as used earlier. This corresponds to diagrams of the form given in Fig. 4. Converting the complex logarithm in I n to a sum of real and imaginary terms via (4), employing the binomial expansion, and replacing the real logarithm by using the replica trick, we arrive at the expression G p,n (y 1 , ..., y p ) − gµ 2 As in Sec. II B we can replace (|φ|/φ) m by a representation that only contains powers of φ. The resulting functional integral can be evaluated, yielding  two-point Green's function in this approximation is then where we have suppressed the spatial argument (y 1 − y 2 ) of G 2 (ε). In the limit of small ε, we recover (49) to linear order and in particular in the double limit, taking δ → 0. This reconfirms the previous results, but also demonstrates that the two-point Green's function has an algebraic δ dependence, given as