Probing the vacuum fluctuations in scalar ghost-free theories

We discuss the response of vacuum fluctuations to a static potential in the context of massive, ghost-free infinite-derivative scalar field theories in two dimensions. For the special case of a $\delta$-like potential, $V=\lambda \delta(x)$, the problem is exactly solvable and we calculate the corresponding Hadamard function for this quantum field. Using this exact result we determine the renormalized value of the vacuum polarization $\langle \hat{\varphi}^2(x)\rangle_\text{ren}$ as a function of the distance $x$ from the position of the potential. This expression depends on the amplitude of the potential as well as the scale of non-locality $\ell$; for distances $x\gg\ell$ the non-local and local results agree, whereas for distances $x<\ell$ there is a difference.


Introduction
The existence of zero-point fluctuations distinguishes a quantum field from a classical field. For a free field in empty flat spacetime these fluctuations are not observable and one usually neglects them. In other words, one considers renormalized quantities in which the contribution of free vacuum zero-point fluctuations is omitted by their subtraction. However, in the presence of matter interacting with the quantum field zero-point fluctuations might lead to observable effects. A famous example is the Casimir effect: The presence of conducting metals and dielectrics changes the propagation of zero-point modes. Their contribution to the vacuum expectation value of the energy is modified, and this energy depends on the shape and position of macroscopic bodies. Thus, as a result of vacuum fluctuations, there appear forces acting on these bodies. This effect was described by Casimir in 1948 [1,2]. In 1997 Lamoreaux [3] directly measured the force between two closely spaced conducting surfaces to within 5% and experimentally confirmed the existence of the Casimir effect.
There are different ways of the calculation of the Casimir force that give the same result [4,5]. Let us consider two parallel conducting pates. As a result of the fluctuations, there exist microscopic currents in the plates. The average of the (retarded) forces between such currents does not vanish and depends on the distance between the plates, and thereby gives rise to the Casimir force. In the other way of calculation, one can focus on the electromagnetic zero-point fluctuations in the cavity between the plates. Taking the presence of the plates into account by properly choosing boundary conditions for the quantum field then yields the Casimir force. In the second approach one can also calculate the renormalized quantum average of stress-energy tensor T µν .
Quantum vacuum averages of quantities that are quadratic in the field depend on the boundary conditions and on an external potential or a current. Quite often, these quantum averages are called vacuum polarization. Using this terminology, one may say that the Casimir effect is a result of the vacuum polarization produced by conducting plates. Certainly, one can characterize the vacuum polarization by considering other quantities instead of the stress-energy tensor. For example, for a scalar fieldφ one may study the properties of φ 2 , and one may consider this object as a "poor man's version of T µν ." In the present paper we use this option. Namely, we consider the fieldφ obeying the following equation In order to specify the operatorD, let us consider an analytic function D(z) of the complex variable z. The operatorD is then obtained by substitution z → − m 2 . We consider and compare two different cases. In both cases V (x) is an external potential producing the vacuum polarization. In the first case we put D(z) = z, such that the operator D is just Klein-Gordon operator − m 2 and Eq. (1.1) describes a local massive scalar field.
In the second case let us instead consider the function D(z) = z exp[f (z)], where f (z) is an entire function (and therefore has no poles in the complex plane). Then, the inverse of this function has only one pole at z = 0. This implies that the propagator 1/D does not have ghosts at tree level and hence the theory (1.1) has the same number of propagating degrees of freedom as in the first case, which is why these theories theories are called ghost-free. Since an exponential of a derivative operator contains infinitely many derivatives by means of its series expansion, these ghost-free theories are also called "infinite-derivative theories" or "non-local theories." We use these terms interchangeably.
Later on, we shall consider a special class of ghost-free theories specified by a positive integer number N , which we call GF N . The parameter is a critical length (or time) at which the modifications connected with the non-locality become important. Technically, this length scale appears in order to form the dimensionless combination 2 ( − m 2 ). Let us introduce the symbol which we call a form factor. These form factors need to have the proper behavior such that we can reproduce the local theory in a certain limit. For this purpose let us consider again the GF N class of theories. In a Fourier basis one has 2 ( − m 2 ) → 2 (ω 2 − q 2 − m 2 ), where ω and q denote the temporal and spatial Fourier frequencies, respectively. The local limit is obtained when ω 1, q 1, and m 1. Hence, in a more general case, it corresponds to the behavior of the differential operatorD(z) at z = 0. Therefore, in order to obtain the correct infrared behavior that reproduces the standard local theory in the limit z → 0, one needs to demand that all physical form factors satisfy α(0) = 1. This is evidently the case for the class of GF N theories (1.3), but there are other choices as well.
Ghost-free field theories, and especially ghost-free gravity has been discussed in a large number of publications, starting from the papers [6][7][8][9]. The main driving force of the study of such theories is an attempt to improve the ultraviolet behavior of the theory without introducing unphysical (ghost) degrees of freedom. For applications of ghost-free gravity for resolving cosmological as well as black hole singularities, see e.g. [10][11][12][13].
The main goal of the present paper is to study the properties of zero-point fluctuations in the ghost-free theory. To probe such fluctuations we consider their response to a specially chosen potential V (x). We restrict ourselves to the simplest case when this potential is static and of the form of a δ-like barrier. We demonstrate that for such a potential both problems, local and non-local one, are exactly solvable. In the main part of the paper we assume that the flat spacetime is two-dimensional. At the end we discuss the higher dimensional versions of the theory, and shall also make remarks on the thermal fluctuations in the ghost-free theory in the presence of the potential V (x).

Scalar ghost-free theory
We begin by considering a simple two-dimensional model of a ghost-free massive scalar field interacting with a potential V . We denote Cartesian coordinates by X = (t, x), such that the Minkowski metric is The action of the theory reads For a quantum fieldφ this action gives the equation (1.1). The operatorD is a function of the Klein-Gordon operator − m 2 . Its explicit form for the local and non-local ghost-free theories was discussed in the Introduction. In order to study the vacuum polarization we choose a static potential V (x) that has a form of a simple δ-function where we assume that this potential is repulsive such that λ > 0. For the calculations we shall employ the formalism of Green functions. Since there exists a wide set of different Green functions related to our problem, let us first discuss them and introduce notations that will be used throughout the rest of this paper.

Green functions "zoo"
In general, we denote a Green function as G(X, X ) with a different choice of the fonts. For the Green functions in the local theory, in presence of the potential, we use the bold font where • = +, −, (1), F, R, A denotes the type of the Green function: The first three objects satisfy the homogeneous equation while the last three objects are solutions of the inhomogeneous equation whereD = − m 2 in this local case. Similarly, in the non-local ghost-free theory the corresponding Green functions (in presence of the potential) are denoted by the bold font version of the calligraphic letters G • (X, X ) . (2.8) These Green functions obey the equations In the absence of the potential, that is, when V (x) = 0, we shall use for the Green functions the same notations, but without bold-facing. The expressions denote free Green functions in the local and ghost-free theories, respectively.
The presence of the potential V (x) breaks the Poincaré invariance of the free theory in two ways: firstly, it violates translational invariance, and secondly, it selects a reference frame in which the potential is at rest. However, since the potential is static, the model preserves the translation invariance in time. This means that all Green functions depend only on the time difference t − t of their arguments. This makes it possible and convenient to use the temporal Fourier transformation. For a function ϕ(t, x) we denote The Fourier transform of the operatorD iŝ (2.14) The temporal Fourier transforms of the above Green functions are marked by the subscript ω: In the presence of the δ-potential the model also has the discrete reflection symmetry x → −x. This implies that

Free local and ghost-free Green functions
Non-local equations are well known in condensed matter theory. For example, the propagation of perturbations in a homogeneous dispersive medium can be described by where is the Laplace operator. Quasiparticles associated with such a theory have the dispersion relation ω 2 = f (−k 2 ), where ω is the energy, and k is a momentum of the quasi-particle. A property which distinguishes the ghost-free theory from other non-local theories is that its action is locally Lorentz invariant. The corresponding dispersion relation is D(−ω 2 + k 2 ) = 0. This means that any solution of the homogeneous equation (1.1) in the local theory is automatically a solution of the homogeneous ghost-free equation. In other words, the on-shell solutions in the local and ghost free case are the same.
Let us present now useful expressions for the temporal Fourier transforms of some Green functions which will be used later. We use the following notations: For this definition both quantities are real non-negative quantities. Let us also notice that in the absence of the potential V the Green functions (both for the local and non-local case) depend only on the difference x − x of their arguments. In what follows we denote this difference simply by x.

(2.22)
Here and in what follows we denote ε ω = sgn(ω). As mentioned previously, all these functions are invariant under the change For ω ≥ 0 the following relation is valid: The last equality is nothing but the fluctuation-dissipation theorem for the vacuum (zero temperature) case, and we shall comment on this in the Conclusion. Let us now discuss the free Green functions for a generic non-local ghost-free theory. 1 Note that the discussion which follows is valid for any non-local theory that can be formulated in terms of one form factor α. To begin with, in the absence of the potential one has The Feynman propagators and the retarded Green functions in the non-local theory differ from their local versions by a universal term ∆G ω (x) as follows: This additional term is given by the integral Since the form factor α has the property α(0) = 1, the integrand is a regular function at q 2 = 2 . Let us also notice that ∆G ω (x) is a real function which is invariant under the transformation x → −x. Lastly, in the local case when α = 1 one has ∆G ω (x) = 0.
In what follows, we will recast all our results in terms of this modification term ∆G ω (x) since it captures the impact of the non-local modification on the local theory.

Green functions in the presence of the potential
In this part we will derive exact expressions for the Hadamard function as well as the causal propagators (retarded and Feynman) for the ghost-free theory in the presence of the δ-potential.

Lippmann-Schwinger equation and its solution
For the calculation of the response of zero-point fluctuations to an external potential one needs to find the corresponding Hadamard Green function. For our choice of the potential it is possible to obtain it in an explicit form. Consider the equation Denote by ϕ 0 ω (x) a solution of the equation for V = 0. Then one can write a solution of (3.1) for the mode function ϕ ω (x) as This is a so-called Lippmann-Schwinger equation [15]. For V (x) = λδ(x) the integral can be taken explicitly and one obtains Here we used that the free Green function G R ω (x, x ) depends only on the difference of the coordinates x − x ; we denote such a function of one variable for x = 0 as G R ω (x). Provided 1 + λG R ω (0) = 0 this algebraic equation can be easily solved and one obtains .

(3.4)
Formally one can employ the free advanced Green function G A ω (x) as well, and it will also solve Eq. (3.1). Expanding a physical wave packet with "advanced modes" instead of "retarded modes" will correspond to different boundary conditions. However, we will prove below that both modes give rise to the same Hadamard function.

Hadamard function
The Hadamard function in the X-representation is defined as the symmetric expectation value . Its temporal Fourier transform is 6) and the symmetry of X ↔ X implies that Using Eq. (3.4) for the field operatorφ ω (x) and the property (2.24) one obtains Let us discuss the properties of this relation. By construction, this expression satisfies (3.7). Secondly, by means of Eq. (2.20), it is proportional to θ(|ω| − m) and hence Lastly, let us notice that This, combined with (3.7), finally implies Again, one might substitute the free advanced Green function G A ω (x) in the above relations. It is related to the free retarded Green function via where we defined the analogous quantity .

(3.13)
Then one may define but using the relations (3.12) as well as (3.10) one sees that Hence, for the calculation of the vacuum polarization in the static case considered here, the retarded and advanced free Green functions can be used interchangeably.

Causal propagators
In this part, let us denote the causal propagators (Feynman and retarded) by the superscript "C." Let us write the causal propagator in the form The solution is given by One may think of this relation as the version of the Lippmann-Schwinger equation for the causal propagators. Again, for V (x) = λδ(x) the above integral can be taken and one finds Combining this relation with (3.16) one gets For x = 0 it reduces to the consistency relation Provided that 1 + λG C ω (0) = 0, we obtain from this algebraic equation the condition .

(3.22)
Therefore one finally obtains for the causal propagators where C=F or C=R for the Feynman or the retarded propagator, respectively. By construction, see Eq. (2.21), the Feynman propagator satisfies as well as The retarded propagator, however, satisfies where the bar denotes complex conjugation.

Interrelation between Hadamard function and causal propagators
Having the exact expressions for the Hadamard function (3.8) as well as the causal propagators (3.23) at our disposal, it is straightforward to show that they are related via Here, G A ω (x, x ) denotes the advanced propagator which can be defined as This implies that also in the X-representation one has (3.31) In particular, one can also show that the Hadamard function G (1) (X, X ) and the Feynman propagator G F (X, X ) are related via which again is due to the Fourier space relation Evidently, similar relations hold for V = 0 and in the local theories. We prove these relations in appendix A. It is important to stress that these interrelations are valid for any non-local modification ∆G ω (x). Ultimately, we are interested in calculating the vacuum polarization which is defined in terms of the Hadamard function. The above relations show that it is also possible to perform the computations using the Feynman propagator, and take the imaginary part only at the end. We will make this more precise in the next section. We are interested in the quantity Inserting (3.8) into (4.1) and using (3.11) one obtains Alternatively, inserting (3.23) into (4.1) as well as making use of the interrelation (3.33) yields The integration limits follow directly from Eqs. (3.9) and (3.26), respectively. At the first glance these two expressions look quite different, but they are, in fact, identical. This can be shown by using the relations detailed in the previous section, as well as in appendix A. Using expression (4.2) it is easy to see that in the absence of the potential barrier, that is, when λ = 0, ϕ 2 (x) ren = 0 as it should be. Using Eq. (2.25) we can isolate the terms encoding the non-locality and obtain (after changing the integration variable from ω to ) the following expression: This is a general expression for the renormalized vacuum polarization for any non-local theory specified by ∆G ω (x) which enters via the dimensionless quantity g ω (x). 2 In what follows, it is our goal to evaluate this expression in the local case, as well as for various non-local cases.

Vacuum polarization in the local theory
Let us first consider the vacuum fluctuations in the local theory which was studied earlier; see [4,17] and references therein. In terms of calculational techniques our approach is quite similar to the one employed in [18]. In what follows we shall use the results of the local theory for the comparison with the results in the ghost-free models. This will allow us to better understand the effects of the non-locality.
In the local case one has ∆G ω (x) = 0, and hence The integral (4.4) then takes the form The scattering of a scalar field on a δ-like potential in a ghost-free theory was studied in [16]. By comparing (4.4) with the results of this paper one can conclude that the factor 1/(1 + C 2 ) which enters the integral (4.4) coincides with the transmission probability R.
Provided m > 0 this integral converges, but it is difficult to evaluate this integral analytically. For x = 0 we can calculate (4.6) analytically and obtain (4.7) where µ := m/λ. Note that ϕ 2 (0) loc. ren is always negative, and asymptotically one has The divergence for µ → 0 corresponds to the well-known IR divergence for a massless scalar field theory in two dimensions.
In the case of x = 0 the vacuum polarization (4.6) can be evaluated numerically. In Fig. 1 we plot the local vacuum polarization ϕ 2 (x) loc.
ren as a function of x for different values of the mass m. For the remainder of this paper we shall focus on GF N non-local theories for which the non-local modification takes the explicit form (4.10) Note that the integrand is manifestly regular at q = for all values of N . It is also clear that for even N the asymptotic behavior in → ∞ is regular, whereas for odd N the asymptotic behavior in is divergent. This feature will become important in the following discussion.

Vacuum polarization in GF 1 theory
The non-local GF 1 theory is defined by the form factor where we defined and erf(z) denotes the error function. In what follows we shall use the fact that the asymptotic of this function for (z)=fixed and (z) → ±∞ is (4.14) From expression (4.12) we can read off where erfi(z) = −i erf(iz) denotes the imaginary error function [19]. Its asymptotic for real z → ∞ is Asymptotically, for finite λ > 0 and ω → ∞, one has Both of these quantities are exponentially divergent for large frequencies . However, the ratio B/C remains finite in this limit: and one has (4.20) The first term in the right-hand side of this expression does not depend on the frequency, and hence the corresponding contribution to ϕ 2 (x) ren is logarithmically divergent. By introducing a UV cutoff Ω one obtains the following expression for the regularized divergent integral: One also has where K 0 (x) is the modified Bessel function. Using these results one can write the expression for ϕ 2 (x) ren in the GF 1 theory as follows: The integral for Ψ(x) is convergent. When adding the Bessel function contribution Z 1 to Ψ(x) we arrive at some "renormalized vacuum polarization" that we can compare to the local expression for ϕ 2 (x) ren . See a graphical comparison of these quantities in Fig. 2.
Our main insights regarding the vacuum polarization in the GF 1 theory are the following: The Gaussian form of the form factor α(z) in this model makes it possible to obtain the Fourier transform of the non-local part of the Green functions (4.12) in an explicit form. This is a very attractive property of this class of ghost-free theories. Namely for this reason, GF 1 theory has been widely used in the study of solutions for static sources. In particular, they effectively regularize the field of a point-like source in four and higher spacetime dimensions (see e.g. [20] and references therein). However, the propagator in this model behaves poorly in the high-frequency regime, resulting in the peculiar behavior of the field created by a time-dependent source in its near zone (see e.g. [21]). In the above calculations of ϕ 2 (x) ren we found that the frequency integral for this quantity is Figure 2. We plot the ϕ 2 (x) ren in the local case as well as in the GF 1 case (where we subtracted the logarithmically divergent term Z 0 ) as a function of the dimensionless distance x/ for a fixed set of potential parameter (λ = 0.5) as well as mass parameter (m = 0.01). At large distance scales, remarkably, the "renormalized vacuum polarization" agrees with the local result. Its shape for small values of x/ is drastically different from the local theory. logarithmically divergent at high frequencies. The origin of this divergence can be easily traced since the integrand in expression (4.10) exponentially grows when → ∞. The same property is valid for any GF 2n+1 theory, wherein the factor in the numerator grows as exp[( ) 2(2n+1) ].
The situation is quite different in the case of GF 2n theories: the corresponding form factor α(z) decreases for both spacelike and timelike momenta when their absolute values tend to infinity. In particular, the integrand in the expression (4.10) exponentially decreases when → ∞ and is of the order of exp[−( ) 4n ]. Thus non-local contributions of GF 2n theories are well-defined and divergence free. However, the analytic calculations in these theories are more involved. In the next section we calculate ϕ 2 (x) ren for the GF 2 theory and show that our expectations regarding the finiteness of the vacuum polarization are correct.

Vacuum polarization in GF 2 theory
The non-local GF 2 theory is defined by the form factor α(z) = exp(− 4 z 2 ) , (4.26) where we introduced the dimensionless quantityx = x. We are not aware of any analytic expression for this integral. This property distinguishes this theory from GF 1 theory and necessitates more involved numerical calculations. It is quite remarkable that for the point at the position of the potential the quantity g ω (0) can be found analytically. One can use the following representation: The integration over the parameter z and then over y leads to the result Let us now consider the case when x = 0. The integrand in (4.27) contains the function f ω (ξ); for small values of it is quite smooth, but for the large value of this parameter is has rather sharp features (see Figure 3). To work numerically, we shall employ a hybrid approach: we approximate the main features of the non-local modification (4.27) analytically and use numerics only for the residual difference between our approximation and the exact expressions (see appendix B for a detailed explanation of our methods).
We find the following large-asymptotics: Here a 2 is a special parameter which we use in our approximation, For more details see Appendix B. Thus one obtains the following asymptotic formulas for the parameters B and C which enter (4.4) valid in the limit of large values : The asymptotics for C can be readily reproduced using an alternative analytical approximation scheme, see Appendix C. As a result we obtain the following asymptotic expression for Φ ω (x) in the limit of large : We see that Φ ω (x) is a decreasing function of . Together with the √ 2 + m 2 -factor in (4.4) the behavior is improved even more. These considerations imply that-unlike in GF 1 theory-the vacuum polarization for GF 2 theory is well-defined and finite for any value of x.
Having a numerical evaluation of g ω (x) at our disposal, we can now numerically evaluate ϕ 2 (x) GF 2 ren . The plot of this function (and the comparison to the local theory) can be found in Fig. 4.
There are a few observations: (i) Asymptotics.-For large distances x the vacuum polarization in GF 2 theory approaches that of the local theory, as expected. As this feature is built into all ghost-free theories considered in this paper, this result confirms that our numerical methods work well.  . Local and non-local vacuum polarization ϕ 2 (x) ren plotted against the dimensionless distance parameter x/ for two different potential parameters (λ = 0.5 and λ = 2). For large distances the local and non-local polarizations approach each other, but for small distance scales x/ ∼ 1 2 there is a crossover between the local and non-local vacuum polarization which we previously discussed elsewhere [22] on a heuristic level. The effect of the non-locality is a smoothing of the polarization around x = 0.
there is a difference between the local theory and GF 2 theory: the vacuum polarization is smoothed out at the origin x = 0 as compared to the local case. This implies that all quantities related to the derivative of the vacuum polarization (∼ ∂ x ϕ 2 ) are now regular at the presence of the δ-potential, whereas in the local theory they are not necessarily continuous. (iii) Overshoot.-Across a wide range of masses and potential parameters (quite possible for all possible values) the vacuum polarization at the location of the δ-potential is numerically larger than in the local case. We call this an "overshoot," and this feature is plotted in Fig. 5. (iv) Crossing.-Lastly, at the intermediate location x ∼ , there is a crossing of the local and GF 2 vacuum polarization. This implies that the difference of the local and non-local vacuum polarization can be both positive and negative. In the GF 1 theory this feature is even more pronounced with multiple crossings, see Fig. 2. We previously discussed these features in the effective energy density in linearized classical nonlocal gravity [22], and it seems that these crossings or oscillations are a generic feature of ghost-free theories.
In the regularized vacuum polarization obtained in the context of GF 1 theory many of these features appear as well, with the notable exception of point no. 3: the vacuum polarization at the location of the potential is more negative than that of the local theory, which we may call "undershoot." Figure 5. We plot the difference of the vacuum polarization at the location of the potential at x = 0 as a function of the potential strength λ . We see that the difference is a function of the dimensionless mass parameter m : for larger masses m at fixed non-locality the difference decreases. In the limiting case λ → 0 the renormalized vacuum polarization vanishes as expected.

Discussion
In this paper we discussed a non-local two-dimensional massive scalar quantum field. For the calculations of the vacuum fluctuations of such a field in the presence of a δ-like potential we employed Green-function techniques.
The calculation of ϕ 2 (x) in the usual local quantum field theory is rather simple. It is greatly simplified by employing a Wick rotation and using the standard methods of the Euclidean theory. In the class of non-local theories which we consider in this paper, however, this method usually does not work: the corresponding form-factor α(z) (see (1.4)) can infinitely grow when its complex argument z reaches infinity along some directions in the complex plane. As a result, one cannot perform a Wick rotation and all the required calculations are to be done in the "physical domain" of the momentum variables. This makes the calculations of the vacuum fluctuations in ghost-free theories much more complicated. In this paper we developed the tools required for these calculations, and this is one of its results.
In order to find ϕ 2 (x) it is sufficient to obtain the Hadamard Green function. We demonstrated that in the absence of the potential the corresponding Hadamard Green function in the ghost-free theory coincides with a similar function in the local theory. We defined ϕ 2 (x) ren as the coincidence limit x → x of the difference of the Hadamard Green function of our model and the free local one. This means that ϕ 2 (x) ren vanishes in the absence of the potential. However, in the presence of the potential, ϕ 2 (x) ren does not vanish in both non-local and local cases, and the corresponding functions depend on the choice of the theory. The second objective of this paper was to study this effect.
In order to simplify calculations we chose the simple model of a repulsive δ-potential. For such a potential one can find the required Green function in an explicit form by solving the field equations by means of the Lippmann-Schwinger method. The expressions for the Hadamard Green function for a general type of the ghost-free theory as well as integral representations for ϕ 2 (x) ren have been obtained in this paper explicitly.
We focused on the calculations of ϕ 2 (x) ren for two ghost-free theories (GF 1 and GF 2 ) and demonstrated that the properties of ϕ 2 (x) ren in these models are quite different. In the GF 1 theory the quantity ϕ 2 (x) ren is logarithmically divergent, whereas in the GF 2 the quantity ϕ 2 (x) ren is a finite smooth function of x for any choice of the mass parameters m and the scale of non-locality . The logarithmic divergence of ϕ 2 (x) ren in the GF 1 theory is an ultraviolet problem connected with the behavior of the GF 1 form factor in the high-frequency domain. In the GF 2 theory (as well as for any GF 2n theory) this problem does not exist. For GF 2 theory we also managed to find an exact analytic expression for ϕ 2 (0) ren at the position of the potential. This provided us with a good test of our numerical computations.
We showed that non-local contributions arise from the universal non-local correction term ∆G ω (x), see Eq. (2.26), which is added to the local causal propagators (retarded, advanced, and Feynman). This correction is real-valued and well-defined in the physical Minkowski space for all GF N theories.
Our numerical computations demonstrated (see Figs. 2 and 4), as we expected, that non-locality smooths the vacuum polarization in the narrow vicinity of the potential and then asymptotically approaches the corresponding value of the local theory. Moreover, at some distance x < , there is a crossover between the local and the non-local vacuum polarization. At the location of the potential the "renormalized" vacuum polarization of GF 1 is more negative than the local polarization, whereas in the completely regular GF 2 vacuum polarization is larger than the local polarization at x = 0.
One might think that the model of a two-dimensional massive scalar field, which we consider in this paper, is over-simplified. However, the methods developed here can be easily generalized and adapted to a more realistic case. Suppose that there exist more than one spatial dimension and denote the coordinates in this space as (x, y ⊥ ). If the potential barrier still has the form λδ(x), one can perform the Fourier transform not only with respect to time t, but also with respect to transverse coordinates y ⊥ . This is possible since the translational invariance into the perpendicular directions is unbroken by the presence of the potential. Denote by k ⊥ the momenta conjugated to y ⊥ . Then one can use the same expression (2.14) for the operatorD ω where now the quantity takes the form Lastly, an additional factor depending on ω appears in the formula (4.3) for ϕ 2 (x) ren , which is connected to the phase volume in momentum space. We hope to address the higher-dimensional problem in a future work.
As a final remark, it would be interesting as well to study the vacuum fluctuations beyond the vacuum state in a thermal bath of finite temperature T . An important connected problem lies in studying under which conditions the fluctuation-dissipation theorem is valid in the class of non-local ghost-free theories. Now we can express all interacting non-local expressions in terms of the free, local Feynman propagator G F ω (x), the real-valued modification ∆G ω (x), and the complex function Λ ω : In the last line we can recast the term proportional to Λ ω Λ ω as follows: Then, we can insert the above expressions into (3.29). Comparing the terms independent of Λ ω as well as linear terms in Λ ω then yields the identity. Realizing that G R ω (x, x )+G A ω (x, x ) is real-valued, one can take the imaginary part of (3.29) and obtain the relation B ∆G ω (x) in GF 2 theory The dimensionless non-local modification g ω (x) = ∆G ω (x) for GF 2 is given by the integral This integral is well-defined, but we are not aware of any analytic solution. For x = 0, however, there exists a solution. Note that in this section, for numerical convenience, we denote f ω (x) as defined in Eq. (4.27) of the main body of the paper as f b (x) instead, where b ≡ ( ) 2 .

B.1 Exact form of ∆G ω (0)
The function f b (ξ) as taken from Eq. (B.1) can be represented as the integral (B.4) Forx = 0 one can take the integrals exactly using the relation Here C and S are the Fresnel integrals. Then We find the asymptotics See a plot of this function in Fig. 7.
B.2 Semi-analytic approach of calculating ∆G ω (x) We are not aware of any analytic solution of (B.1) for x = 0. In what follows, we will describe our method for (i) evaluating this integral numerically and (ii) extracting the asymptotic behavior for large . Depending on the value of the dimensionless parameter b, the function f b (ξ) takes rather different shapes, see Fig 3. Calculating the extrema of f b (ξ) we find a local maximum at ξ = 0 and a minimum at ξ = ξ + . Moreover, provided b is large enough, there is another local maximum at ξ = ξ − : where E < b 0 ,ξ∞ (x) denotes the error of this approximation. On the other hand, for the regime b > b 0 it is useful to approximate the peak around ξ = 1 analytically. The following approximation works well: where we defined Ei(z) := Ei(z) for (z) ≥ 0 , Ei(z) + iπ for (z) < 0 , (B.23) which implements the branch cut of the exponential integral for arguments with negative real part.

B.3 Asymptotics x = 0
Using the approximation presented in Eq. (B.11) we can extract the behaviour of (B.1) for large values of b and find (for fixedx) the rather crude approximation The above implies that g ω (x) behaves as an oscillatory term of magnitude 1 2 for large values of . In the above, we made use of the relations (B.25) See Fig. 7 for a graphic confirmation of these asymptotics.  C Remarks on ϕ 2 (x) ren in GF 2n for larger n As it turns out it is possible to analyze the large− asymptotics of g ω (0) in the case of GF 2n theories for any n, which we will describe in the following. Consider again the expression