PT-symmetric quantum field theory in D dimensions

PT-symmetric quantum mechanics began with a study of the Hamiltonian $H=p^2+x^2(ix)^\varepsilon$. A surprising feature of this non-Hermitian Hamiltonian is that its eigenvalues are discrete, real, and positive when $\varepsilon\geq0$. This paper examines the corresponding quantum-field-theoretic Hamiltonian $H=\frac{1}{2}(\nabla\phi)^2+\frac{1}{2}\phi^2(i\phi)^\varepsilon$ in $D$-dimensional spacetime, where $\phi$ is a pseudoscalar field. It is shown how to calculate the Green's functions as series in powers of $\varepsilon$ directly from the Euclidean partition function. Exact finite expressions for the vacuum energy density, all of the connected $n$-point Green's functions, and the renormalized mass to order $\varepsilon$ are derived for $0\leq D<2$. For $D\geq2$ the one-point Green's function and the renormalized mass are divergent, but perturbative renormalization can be performed. The remarkable spectral properties of PT-symmetric quantum mechanics appear to persist in PT-symmetric quantum field theory.


I. INTRODUCTION
The study of PT -symmetric quantum theory may be traced back to a series of papers that proposed a new perturbative approach to scalar quantum field theory. Instead of a conventional perturbation expansion in powers of a coupling constant, it was proposed that a parameter δ that measures the nonlinearity of the theory could be used as a perturbation parameter [1,2]. Thus, to solve a gϕ 4 field theory we studied a gϕ 2 ðϕ 2 Þ δ theory and treated the parameter δ as small. The procedure was to obtain a perturbation expansion in powers of δ and then to set δ ¼ 1 to obtain the results for the gϕ 4 theory. Detailed investigation showed that this perturbative calculation is numerically accurate and does not require the coupling constant g to be small [1,2]. An important feature of this approach was that ϕ 2 and not ϕ had to be raised to the power δ in order to avoid raising a negative quantity to a noninteger power, thereby generating complex numbers as an artifact of the procedure.
PT -symmetric quantum mechanics began with the surprising discovery that spurious complex numbers do not appear if the quantity raised to the power δ is PT symmetric (invariant under combined space and time reflection) [4,5]. This fact is highly nontrivial and was totally unexpected. Indeed, the eigenvalues of the non-Hermitian PT -symmetric Hamiltonian are entirely real, positive, and discrete when ε ≥ 0 because ix is PT invariant. A proof that the spectrum is real when ε > 0 was given by Dorey, Dunning, and Tateo [6,7].
The purpose of this paper is to introduce powerful new tools and techniques that can be used to investigate PT -symmetric quantum field theories. We illustrate these tools by studying the quantum-field-theoretic analog of (1) whose D-dimensional Euclidean-space Lagrangian density is where ϕ is a pseudoscalar field so that L is PT invariant. We treat ε as small and show how to calculate the vacuum energy density E 0 , the connected n-point Green's functions G n , and the renormalized mass M R as series in powers of ε.
In this paper we assume that 0 ≤ D < 2 to avoid the appearance of renormalization infinities and then we comment briefly on the perturbative renormalization procedure for the case D ≥ 2.
To first order in ε (ε ≪ 1), the unusual Lagrangian density L in (1) has a logarithmic self-interaction term: For a quantum field theory having a complex logarithmic interaction term it is not obvious whether one can find Feynman rules for performing perturbative diagramatic calculations. We will show how to construct such Feynman rules. We begin by replacing the complex logarithm with a real logarithm and we do so in such a way as to preserve PT symmetry; to wit, we define logðiϕÞ ≡ 1 2 iπ þ logðϕÞ ðif ϕ > 0Þ and we define logðiϕÞ ≡ − 1 2 iπ þ logð−ϕÞ ðif ϕ < 0Þ: Combining these two equations, we make the replacement Note that in (4) the imaginary part is odd in ϕ and the real part is even in ϕ. Thus, (4) enforces the PT symmetry because the pseudoscalar field ϕ changes sign under parity P and i changes sign under time reversal T . [To derive (4) we must assume that ϕ is real. The reality of ϕ is explained in Sec. II.] Graphical techniques were developed in Ref. [1] to handle real logarithmic interaction terms. These techniques are generalizations of the replica trick [22], which has been used in the study of spin glasses. The idea of the replica trick is that a logarithmic term log A can be reformulated as the limit log A ¼ lim N→0 1 N ðA N − 1Þ, or equivalently and slightly more simply as the limit One then regards N as an integer and identifies A N as an N-point vertex in a graphical expansion. Of course, this procedure is not rigorous because it requires taking the continuous limit N → 0. The validity of this approach has not been proved, but when it is possible to compare with exactly known results in low-dimensional theories, the replica trick gives the correct answer. In this paper we verify our field-theoretic results by comparing them with the exact answers for D ¼ 0 (where the functional integral becomes an ordinary integral) and for D ¼ 1 (quantum mechanics). The graphical calculations in this paper are done in coordinate space. Once the vertices have been identified, all that one needs is the free propagator in D-dimensional Euclidean space Δðx − yÞ, which satisfies the differential equation Taking the Fourier transform of this equation gives the amplitude for the free propagator of a particle of mass 1 in momentum space:Δ The D-dimensional inverse Fourier transform of this expression then gives the D-dimensional coordinate-space propagator in terms of an associated Bessel function: If we let x 1 → x 2 , we obtain the amplitude Δð0Þ for a self loop, which is the amplitude for a line to originate from and return to the same point: This expression is finite and nonsingular for 0 ≤ D < 2. This paper is organized as follows. We calculate the ground-state energy density E 0 in Sec. II, the one-point Green's function G 1 in Sec. III, the two-point Green's function G 2 and the renormalized mass M R in Sec. IV, and the general connected n-point Green's function G n in Sec. V, all to first order in ε. These quantities are finite when 0 ≤ D < 2, but the three quantities E 0 , G 1 , and M R diverge when D ≥ 2 so it is necessary to introduce a renormalization procedure. In Sec. VI we discuss the issues of renormalization. We show that a redefinition of the energy scale, an additive shift in the field, and a mass counterterm eliminate these infinities. In this section we also discuss our future calculational objectives, namely, calculating the Green's functions to higher order in ε.

II. FIRST-ORDER CALCULATION OF THE GROUND-STATE ENERGY DENSITY
If we expand the partition function for the Lagrangian density L to first order in ε and use (4), the functional integral (9) becomes where the free Lagrangian density is obtained by setting ε ¼ 0 in L. Note that the imaginary part of the functional integrand in (10) is odd in ϕ, so ZðεÞ is real.
We emphasize that the functional integration in (10) is performed along the real-ϕ axis and not in the complex-ϕ domain. This justifies the use of (4). We are not concerned here with complex functional integration paths that terminate in complex Stokes sectors because the functional integral (10) converges term-by-term in powers of ε. In PT -symmetric quantum mechanics the boundary conditions on the Schrödinger equation associated with the Hamiltonian (1) [see (A2)] are imposed in complex Stokes sectors [5]. However, in the context of quantum field theory it would be hopelessly unwieldy to consider functional Stokes sectors. This is why we treat ε as small. This paper is concerned with calculating the coefficients in the ε series and we do not consider here the mathematical issues involved with the summation of such a series for large values of ε.
In general, a partition function is the exponential of the ground-state energy density E 0 multiplied by the volume of spacetime V: Z ¼ e −E 0 V . Thus, the shift in the ground-state energy density ΔE to order ε is given by Hence, from (5) we obtain This expression has a graphical interpretation as the product of N self loops from the spacetime point y back to y. There are exactly ð2N − 1Þ!! ways to construct these self loops, so the expression for ΔE simplifies to Next, we note that the integral R d D y is the volume of spacetime V, so this formula simplifies further: Finally, we use the duplication formula for the gamma function [23] to write and then take the derivative with respect to N to get The result in (12) may be verified for the special cases of D ¼ 0 and D ¼ 1.
Special case D ¼ 0: For D ¼ 0 the normalized partition function becomes an ordinary integral, which we can expand to first order in ε: where Z 0 ¼ ffiffiffiffiffi ffi 2π p . If we take the negative logarithm of this result, we obtain the first-order shift in the ground-state energy density We then integrate separately from −∞ to 0 and from 0 to ∞ and combine the two integrals to obtain a single real integral that we evaluate as follows: Taking D ¼ 0 in (8) gives Δð0Þ ¼ 1, so (12) reduces exactly to (14) in zero-dimensional spacetime.
Note that in PT -symmetric quantum mechanics the calculation of expectation values requires the C operator [8]. However, the C operator is not needed for any of the calculations in this paper because a functional integral involves vacuum expectation values. The vacuum state is an eigenstate of the C operator with eigenvalue 1, Cj0i ¼ j0i, so all reference to C disappears. This simplification was first pointed out in Ref. [24].

III. FIRST-ORDER CALCULATION OF THE ONE-POINT GREEN'S FUNCTION
The one-point Green's function G 1 is nonperturbative in character but it can be calculated by following the approach used above to calculate ΔE. Keeping terms that do not vanish under ϕ → −ϕ, we evaluate directly the functionalintegral representation where L 0 is the free Euclidean Lagrangian in (11). We use the integral identity to replace ϕðyÞjϕðyÞj in the functional integral (16) and then we replace sinðtϕÞ by its Taylor series [25]: This converts (16) to the product of an infinite sum in n, a one-dimensional integral in t, a D-dimensional integral in y, and a functional integral in ϕ: The sum and multiple integrals in (19) may appear to be difficult, but like the calculation of the ground-state energy density, the functional integral in the second line of (19) also has a graphical interpretation; it is merely the product of the free propagator Δða − yÞ representing a line from a to y multiplied by n þ 1 self loops from y to y, and this product is accompanied by the combinatorial factor ð2n þ 3Þ!!. Thus, the second line in (19) reduces to ð2n þ 3Þ!!Δðy − aÞΔ nþ1 ð0Þ.
This result simplifies further because, as we can see from (6), the D-dimensional integral is trivial: R d D yΔðy−aÞ¼1. This establishes the translation invariance of G 1 . The rest is straightforward: This expression for G 1 is exact to order ε.
Observe that the expression for G 1 is a negative imaginary number. This is precisely what we would expect based on previous studies of classical PT -symmetric systems. The classical trajectories in complex coordinate space of a particle described by the Hamiltonian (1) are leftright symmetric but they lie mostly in the lower-half plane [8]. Thus, the average value of the classical orbits is a negative-imaginary number.
Special case D ¼ 0: The one-point Green's function in D ¼ 0 is given by the second line in (13) with an additional extra factor of ϕ: where Zð0Þ ¼ ffiffiffiffiffi ffi 2π p . The integration over the first term in the square brackets vanishes by oddness. We evaluate the contribution of the second term by integrating first from −∞ to 0 and then from 0 to ∞. Combining these two integrals, we obtain This result is in exact agreement with the general result in (20) because Δð0Þ ¼ 1 when D ¼ 0.
Special case D ¼ 1: When D ¼ 1, the expression for G 1 in (20) reduces to In the Appendix we derive G 1 in quantum mechanics and verify (21).

IV. FIRST-ORDER CALCULATION OF THE TWO-POINT GREEN'S FUNCTION
To obtain the connected two-point Green's function G 2 ða; bÞ, one must subtract G 2 1 from the vacuum expectation value of ϕðaÞϕðbÞ. However, we have seen that G 1 is of order ε. Therefore, to first order in ε we need only evaluate ZðεÞ in (10) with ϕðaÞϕðbÞ inserted after Dϕ and then divide this integral by ZðεÞ. We may neglect the imaginary terms in these integrals because they are odd under ϕ → −ϕ. Thus, the expression that we must evaluate for G 2 ða; bÞ is We expand this expression to first order in ε as a sum of three functional integrals: We must now evaluate the three contributions A, B, and C. The functional integral A in (23) is simply the free propagator Δða − bÞ. This result verifies that if we set ε ¼ 0 in the Lagrangian (2), we obtain a free field theory; the two-point Green's function for such a field theory is Δða − bÞ.
The double functional integral C presents a complication. The first line of C is proportional to A and evaluates to 1 4 εΔða − bÞ. However, as we showed in the calculation of the ground-state energy, the next two lines of C evaluate to 4 ε VΔE, where ΔE is the first-order shift in the ground-state energy density (12) and V is the volume of Euclidean spacetime. Thus, C ¼ Δða − bÞVΔE, and this quantity is divergent because V is infinite.
We resolve this divergence problem by calculating B. Using (5) we express the B integral as This functional integral requires that we connect in a pairwise fashion the set of 2N þ 2 points consisting of a, b, and the 2N points y with the free propagator Δ in (7). There are two cases to consider. In the first case a is connected to b and the remaining 2N points at y are connected in pairs. Note that this reproduces the result above for C except with the opposite sign. Thus, the volume divergences exactly cancel.
In the second case a is not connected to b. Instead, a connects to a point y (there are 2N ways to do this) and b connects to one of the remaining 2N − 1 points y (there are 2N − 1 ways to do this). The rest of the 2N − 2 points y are joined in pairs [there are ð2N − 3Þ!! ways to do this]. The amplitude for this case is Thus, our final result for the coordinate-space two-point Green's function to order ε is In momentum space this is From (27) we construct the (0, 1) Padé approximant, which is just the geometric sum of a chain of bubbles: We then read off the square of the renormalized mass to first order in ε: Special case D ¼ 0: To verify (28) and (29) for the case D ¼ 0 we evaluate the ordinary one-dimensional integrals in These integrals are not difficult and to order ε we get This result agrees exactly with that in (28) for D ¼ 0.
Special case D ¼ 1: We verify (29) for the case D ¼ 1 by calculating the energy level of the first excited state of the Hamiltonian H in (1). When ε ¼ 0, H becomes the harmonic-oscillator Hamiltonian, so the first excited state eigenfunction is xe −x 2 =2 and the associated eigenvalue is 3 2 . We solve the time-independent Schrödinger equation Hψ ¼ Eψ perturbatively by substituting and collecting powers of ε. To first order in ε the function ψ 1 ðxÞ satisfies To solve this equation we use the technique of reduction of order and let ψ 1 ðxÞ ¼ xe −x 2 =2 fðxÞ. The equation for fðxÞ is then xf 00 ðxÞ þ ð2 − 2x 2 ÞfðxÞ ¼ −2E 1 x þ x 3 logðixÞ:

Multiplying this equation by the integrating factor
Therefore, if we integrate this equation from −∞ to ∞, we obtain an equation for E 1 : where we have replaced logðixÞ by 1 2 logðx 2 Þ. These integrals are easy to evaluate and we get Thus, the first excited eigenvalue of H to order ε is We have already calculated the ground-state energy in Sec. II: The renormalized mass M R is the first excitation above the ground state so M R is the difference of these two energies: If we square this result and keep terms of order ε, we get which exactly reproduces (29) for the case D ¼ 1.

V. HIGHER-ORDER GREEN'S FUNCTIONS
The connected three-point Green's function is given by the cumulant However, to order ε The calculation of G 3 is somewhat tedious, but the procedure follows exactly the calculation of the two-point Green's function. The final result after the disconnected terms have canceled is The connected four-point function is defined by the cumulant Again, calculating G 4 is tedious but the result is simply The pattern is now evident and with some effort we can calculate the connected n-point Green's function to order ε and obtain a general formula valid for all n ≠ 2: Thus, the connected Green's functions are all of order ε except for G 2 in (26), which is of order 1. Observe that (38) reduces to (20), (36), and (37) for n ¼ 1, 3, and 4. We emphasize that (38) holds for both odd and even n. Our calculation of the odd-n Green's functions uses the first term on the right side of (4) and our calculation proceeds by introducing the integral representation in (17) followed by using the Taylor series in (18). On the other hand, our calculation of the even-n Green's functions uses the second term on the right side of (4) and the calculation proceeds by applying the derivative identity in (5). It is satisfying that these two strikingly different techniques lead to the single universal formula in (38) for G n .

VI. DISCUSSION AND FUTURE WORK
The principal advance reported in this paper is that we have developed all of the machinery necessary to calculate the Green's functions of a PT -symmetric quantum field theory in (2) to any order in ε. Thus, this paper opens a vast area for future study and investigation; one can investigate the masses of the theory (the poles of the Green's functions), scattering amplitudes, critical indices, and so on. The Green's-function calculations done in Secs. II-V are exact to first order in ε. However, the procedures presented here immediately generalize to all orders in ε.
Furthermore, as we show below, even in low orders the perturbation series in powers of ε is highly accurate and it continues to be accurate for large ε. To illustrate, we calculate the one-point Green's function G 1 in D ¼ 0 to second order in ε. In D ¼ 0 this Green's function is a ratio of two ordinary integrals: where L ¼ logðixÞ ¼ 1 2 iπjxj=x þ 1 2 logðx 2 Þ. Evaluating these integrals is straightforward and the result is To check of the accuracy of (40) we calculate the onepoint Green's function for a cubic theory (ε ¼ 1). We convert the expansion in (40) to a [0, 1] Padé approximant, and then set ε ¼ 1 to obtain the approximate result The exact value of G 1 for the zero-dimensional cubic theory ε ¼ 1 is given by the ratio of integrals

Γð2=3Þ
Γð1=3Þ Thus, the two-term ε expansion (40) has an accuracy of 5%, which is impressive for such a large value of ε. This good result is consistent with the results found in previous studies of the accuracy of the ε expansion for various classical equations (see Ref. [3]). The third-order version of (40) is Converting the expansion above to a [1,1] Padé approximant and setting ε ¼ 1, we obtain the result that G 1 ¼ −0.6213i, which now differs from the exact result by only −2%. These numerical results strongly motivate us to extend our studies of the ε expansion of PT -symmetric quantum field theories to higher order in ε. We will publish the higher-order calculations in a future paper. A second issue that that needs to be examined in depth is that of renormalization. Because Δð0Þ becomes singular when the dimension D of Euclidean spacetime reaches 2, the one-point Green's function G 1 and the renormalized mass M R become singular. (To first order in ε the higher Green's functions do not become infinite when D ¼ 2.) Thus, for D ≥ 2 we must undertake a perturbative renormalization procedure.
For simplicity, in this paper we have worked entirely with dimensionless quantities. However, to carry out a perturbative renormalization one must work with the Lagrangian ϕÞ ε for which the dimensional parameters are explicit: μ is the unrenormalized mass μ, μ 0 is a fixed parameter having dimensions of mass, and g is a dimensionless unrenormalized coupling constant. The mass renormalization procedure consists of expressing the renormalized mass M R in terms of these Lagrangian parameters and absorbing the divergence that arises when D ≥ 2 into parameter μ. The coupling-constant renormalization procedure is similar; we define the renormalized coupling constant G R as the value of the three-point or four-point Green's functions at particular values of the external momentum and again absorb the divergence that arises into the Lagrangian parameter g. One must then verify that all higher Green's functions are finite when expressed in terms of M R and G R . This program will be carried out explicitly in a future paper.

ACKNOWLEDGMENTS
C. M. B. thanks the Heidelberg Graduate School of Fundamental Physics for its hospitality.

APPENDIX
In this appendix we calculate the one-point Green's function G 1 for the case D ¼ 1 (quantum mechanics). In quantum mechanics G 1 is the expectation value of the operator x in the ground state j0i. Thus, if ψðxÞ is the (unnormalized) ground-state eigenfunction in coordinate space, we can express G 1 as the ratio of integrals For the quantum-mechanical Hamiltonian (1) the ground-state eigenfunction obeys the time-independent Schrödinger equation where E is the ground-state energy. To first-order in ε this differential equation becomes − 1 2 ψ 00 ðxÞ þ 1 2 x 2 ψðxÞ þ 1 2 εx 2 logðixÞψðxÞ ¼ EψðxÞ: ðA3Þ We solve this equation perturbatively by substituting ψðxÞ ¼ ψ 0 ðxÞ þ εψ 1 ðxÞ þ Oðε 2 Þ; where ψ 0 ðxÞ ¼ expð− 1 2 x 2 Þ, E 0 ¼ 1 2 , and as derived in (15), Collecting powers in ε, we see that ψ 0 ðxÞ and ψ 1 ðxÞ satisfy the differential equations We then multiply this equation by the integrating factor expð−x 2 Þ and integrate from −∞ to x: The integration constant C vanishes because Thus, we find that We can now evaluate the integrals in (A1) to first order in ε. This expression simplifies considerably because of the factor of x: To evaluate this integral we integrate from x ¼ −∞ to 0 and then from x ¼ 0 to ∞ and combine the two integrals. The resulting integral simplifies further because the logarithm in the integrand collapses to logð−1Þ ¼ iπ: Evaluating this triple integral is not trivial, but by using some tricks it can be calculated exactly and in closed form. We begin by interchanging the order of the s and t integrals: Next, we introduce polar coordinates x ¼ r cos θ and s ¼ r sin θ and make the change of variable t ¼ rz: The r integral can now be done, and we obtain a sum of two double integrals: These double integrals may be evaluated by using any algebraic manipulation code such as Mathematica. The final result is This verifies the general result in (20) for the case D ¼ 1.