Nonlocality of quantum matter corrections and cosmological constant running

Semiclassical contributions to the gravitational action include various terms with zero, two, and four derivatives of the metric as well as nonlocal form factors for these terms. Contributions to some of these terms could be confused with others on a specific metric background or for a particular gauge fixing. We present a critical analysis of the recent works where the tensor structure and the number of derivatives in the action of gravity were not properly taken into account. Taking these relevant aspects into account, we show that although some contributions owing to the quantum fluctuations of massive or massless scalar as well as fermion and vector fields may be attributed to the"running"of the cosmological constant, in reality they correspond to the fourth derivative terms of the action.


Introduction
Quantum matter corrections to the effective gravitational action is one of the main points of interest in semiclassical gravity staring from the seminal pioneering works [1,2]. A characteristic feature of the contributions of massive quantum field is the presence of more complicated nonlocal form factors in the vacuum effective action [3,4,5,6] 1 that boil down to the logarithmic expressions in the UV (at large energies) [7,3]. The references [3,4,5,6,8] employ the heatkernel technique providing one of the most direct and unambiguous methods to derive the mentioned form factors. An alternative equivalent approach is to use Feynman diagrams on the flat Minkowski spacetime background η µν [5,9]. Decomposing the metric as follows one has to calculate the self-energy diagrams (or the polarization operator [2]) for the graviton propagator G µναβ = −i h µν h αβ shown in Fig. 1 and then determine the semiclassical corrections to the gravitational action. The role of mass for the problem under consideration is crucial. At low energy (in the IR), in the momentum representation, one can make an expansion in the ratio k 2 /m 2 (where k 2 = k µ k µ is the square of Euclidean momentum) and arrive at the gravitational analog of the Appelquist and Carazzone decoupling theorem [10]. This decoupling has been found in [5], but only for the C 2 and R 2 terms, where C µναβ is the Weyl tensor. In a recent work [8], one can see the decoupling taking place for the Einstein-Hilbert term, but not for the cosmological constant term.
One of the interesting outputs of calculations [5] is that they do not provide a nonlocal form factor for the cosmological constant term. The reason is that the hypothetical nonlocal form favor k Λ ( ) should act on a constant term, giving zero. However, it was suggested in [5] and in a slightly different framework in [11] that an effective running of the cosmological term ρ Λ would be possible if we meet, in the quantum corrections to the gravitational action, the logarithm form factors in the terms quadratic in the Ricci tensor or the curvature scalar, or in a similar term with the Riemann tensor, or in higher order terms such as F ( −1 R), etc. To see the reason why terms (2) are nonlocal analogs of the cosmological constant term (called the CC term, in what follows), one has to consider a local (similar to the cosmological evolution) scaling of the metric where η is the conformal time. For a = const, terms (2) scale exactly like the CC term. Euristically, the d Alembert operator in denominator is traded for two derivatives in R µν or R. For a time-dependent a(η), these nonlocal terms provide typically a mild deviation from the CC term and this explains the phenomenological success of the corresponding cosmological models [12] (see also further references therein). In general, cosmological models with slowly varying CC term attract a lot of attention and for this reason, it would be very interesting to derive the nonlocal terms (2) explicitly. However, the attempts to do so, failed so far. In particular, (a) (b) Figure 1: Two diagrams of the one-loop correction to h µν h αβ due to a quantum matter field represented by thick lines. Diagram (a) gives a contribution to the nonlocal form factor, while diagram (b) contributes only to the local expression, i.e., divergences. Both plots are needed to provide a correspondence between logarithmic divergences and the logarithmic UV limit for the form factors [5].
the nonlocal terms that emerge in the vacuum action from a spontaneous symmetry breaking (SSB) [13] (see also more detailed consideration, including highly non-trivial derivation of the energy-momentum tensor of vacuum, in [14]) produced a negative result. Although there are many nonlocal terms in the SSB-induced action, the ones of type (2) cancel out. In this situation, it looks very promising that similar terms with extra logarithm factor were found in a recent work [15]. According to [15], these expressions were obtained from the very same diagrams in Fig. 1 calculated in [5,9] and this fact requires, in view of the above mentioned, an explanation. An alternative description of quantum matter effects is the renormalization group running. The physical beta functions can be derived using the momentum-subtraction scheme of renormalization. In the IR, these beta functions demonstrate the quadratic decoupling, however, only for the coefficients of the C 2 , R 2 terms [5], and the R-term [8].
It is worth noting that in the case of the minimal subtraction (MS) renormalization scheme, there is a well-defined beta function for ρ Λ [16,17]. An interesting paper [18] entitled "Cosmology is not a renormalization group flow" compares the MS-scheme beta function for ρ Λ with the explicit calculation of quantum contributions to the vacuum energy on de Sitter space and finds different results. However, this comparison was made for the quantum effects of a massless scalar field. On the other hand, the beta function for ρ Λ vanishes in the MS or any other scheme of renormalization for all known types of massless fields. Thus, one has to explain the discrepancy between the vanishing beta function and the nonzero result on a de Sitter background.
It turns out that the explanation of the two results concerning the quantum contributions to the vacuum energy density ρ Λ [15] and [18], is quite similar, regardless that they were obtained by completely different methods. In both cases, subtleties in the interpretation of the calculations are connected with the tensor structure of the gravitational action. We believe that explaining these results in more detail will be instructive and present the corresponding explanations in the rest of this paper.
The work is organized as follows. Section 2 briefly reviews the expression for the nonlocal effective action derived in [5]. In Sec. 3, we discuss quantum matter corrections in de Sitter space and in Sec. 4 provide an in detail analysis of the results of the calculation of Feynman diagrams in [15]. Finally, we draw our conclusions in Sec. 5.

Form factors for the gravitational action
Let us start by introducing some standard notions. The general expression for the action of semiclassical gravity providing renormalizable semiclassical theory, has the form S vacuum = S EH + S HD (6) and consists of the Einstein-Hilbert term and the higher derivative term which includes the square of the Weyl tensor C 2 = R 2 µναβ − 2R 2 αβ + 1/3 R 2 and the integrand of the Gauss-Bonnet topological invariant E 4 = R µναβ R µναβ − 4 R αβ R αβ + R 2 . Here a 1 ,..., a 4 , G , and Λ are the parameters of the semiclassical gravitational action. As in [5] and [15], we consider quantum matter corrections due to a free massive scalar field with the nonminimal parameter of interaction ξ. The one-loop contributionΓ vac to the gravitational action was obtained in [5] by using the diagram approach with the expansion in the curvature tensor and its covariant derivatives up to the second order in curvature (and verified employing the heat kernel method). The result of these calculations has the form with nonlocal form factors In these formulas, we consider Euclidean signature and use the notations Two things are worth mentioning here.
i) The one-loop contribution (10) is complete in the O(R 2 ... ) approximation. ii) In the massless case, there is no m 4 -type counterterm and the MS-scheme beta function for ρ Λ vanishes. Let us note, in passing, that in the massless case there are no corrections to the CC term also at higher loops. In both massive and massless cases, there is no nonlocal form factor for the CC term. Only the local counterterm ∝ m 4 is present in the massive case.
Results i) and ii) are in a direct conflict with the interpretations of [18] and [15], where the quantum correction to the CC cosmological constant equation of state ω = −1 in the massless theory and the m 4 -type nonlocal one-loop correction to the CC were found, respectively. We clarify an apparent contradiction of the obtained results in the sections below and show that all results are consistent taking into account that the loop contributions to the vacuum action correspond not only to the ρ Λ -term but, also, to other relevant terms in the classical action (6). However, only terms (8) have logarithmic divergences and, therefore, only these terms gain logarithm ( -dependent) form factors from the loop corrections.

IR dependence on a in the massless case
The logic of [18] is based on the comparison of the renormalization group equation for ρ Λ in [19], [20], and other papers on the renormalization group running of the CC and the quantum corrections to the vacuum equation of state obtained in [21]. The last calculation is based on the well-defined stochastic quantization of massive scalar field on a de Sitter background [22]. This technique enables one to obtain higher-loop perturbative and even nonperturbative results. 2 The massless limit in this method is smooth and one should expect a good fit with other methods of calculations.
The first-order logarithm corrections for the energy density and the pressure of vacuum, quoted in [18], have the form 2 Compared to the recent attempt to apply the stochastic formalism to the case of an arbitrary metric background [23], using the de Sitter metric provides significant advantages. As we will show below, this also means the need for a special care in the interpretation of results.
One can identify these terms as two-loop contributions because they are proportional to the coupling constant λ of the ϕ 4 interaction. At one loop, it is very well known that the vacuum contributions do not have such a dependence. On the other hand, in a usual perturbative treatment, there are ln (a)-terms but only in the effective action and not in the elements of the energy-momentum tensor, such as ρ vac and p vac . Let us start the discussion by saying that we do not contest the correctness of the calculations leading to (13). On the other hand, we believe that these calculations should be correctly interpreted. It looks misleading to compare them with the running of the cosmological constant term, described in different frameworks (e.g., in [24] or [19]).
First, we can pose a question which may be useful as a starting point. Since the oneloop contribution to the effective action (10) has nor CC-type divergences, neither nonlocal corrections to the CC term in the massless case, how one can resolve the contradiction between (13) and (10) in the limit m → 0?
Let us recall that the calculations of [21] were done on a de Sitter background, where the terms in action (6) boil down to the expressions Taking this into account and looking at Eq. (13), we readily see that the logarithmic "running" should not be attributed to the CC term ρ Λ , but rather to a linear combination of E 4 = 24H 4 and R 2 = 144H 4 . Indeed, when making the comparison of the running of the parameters in action (10) with the output of de Sitter calculations, the unique reasonable attribution is to the running of a 2 and a 3 in Eq. (8) and not of the CC term. The reason is (as we mentioned above) that the constant ρ Λ does not run in massless models and, also, it is not O(H 4 ), while the E 4 and R 2 -terms are exactly of order H 4 One can note that, in the conformal (ξ = 1/6) massless case, the corrections to ρ and p can be easily recovered from the trace anomaly [25,26] and the conservation equation, as described in [27] and, more recently, in [28] (see further references therein). Unfortunately, our attempt to recover the coefficients in (13) from the known one-loop beta functions did not work well, probably because of the second loop "contamination". Another relevant observation is that, in the higher derivative model [19], there are both massless and massive modes, but it is not clear how to separate them in the IR and arrive at the meaningful comparison with the IR evolution described in [21] in the massless case.
It would be certainly interesting to find an analog of the formulas (13), but this is not an easy task. Let us start from a simpler problem and present the expressions for the one-loop four-derivative contributions on de Sitter space for a conformal scalar field. In this case, the trace anomaly has the form T µ µ = ωC 2 + bE 4 + c✷R, One can make a comparison between the anomaly and the anomaly-induced action, which can be presented in the simplest form using the variablesḡ µν and σ = ln (a), where The standard non-covariant form of the induced action is In the case of de Sitter background, there is a term bσĒ 4 , which becomes 24H 4 log(a) on de Sitter background.
In the cosmological setting, e.g., assuming a conformally flat FRW metric g µν = a 2 (η)η µν , one can combine the conservation law and the relation for the trace, i.e., to arrive at the one-loop contributions to the energy density and pressure of the vacuum [27,28], where the prime stands for the derivative with respect to conformal time η. On the de Sitter background, with H = a ′ /a 2 = const, this boils down tō Compared to the formulas (13), we can see that the relations (20) nicely reproduce the general factors H 4 . This is explained by the fact that (13) should not be interpreted as a correction to the cosmological constant term, but to the fourth derivative terms. On the other hand, there are the following serious differences between (20) and (13): i) The relations (20) do not depend on the coupling constant λ, which is present in (13). This shows that the last formula is a two-loop contribution, while the former comes from the one-loop contribution.
ii) Different from the effective action (17), the relations (20) do not have the logarithms of a, which are present in (13). This is certainly related to the previous point. At the second loop, usually there are terms quadratic in the logarithms which should produce ln (a) in thē ρ dS vac andp dS vac . However, in the massless theory, these logarithms describe the fourth derivative terms and not the one of the cosmological constant term.
iii) The formulas (20) correspond to the conformal scalar, while (13) correspond to the minimal scalar. This difference is not critical and can be eliminated by assuming the non-local running of the coefficient of the R 2 -term in the minimal theory. A pertinent observation is in order in this respect. The minimal interacting scalar theory is not renormalizable even at the one-loop level, different from the conformal scalar theory. For this reason, the vacuum effective action in the minimal version of the scalar theory is plagued by severe non-renormalizable divergences at the two loop order. This maybe not a problem for stochastic formalism, on which the derivation of (13) was based. However, this point makes any kind of comparison with the usual perturbative formalism a completely non-trivial issue.
As we mentioned above, the expression (17) represents a local version of the renormalization group running which, in difference to a global scaling (as correctly noted in [18]), can be consistently applied to cosmology. However, this action is well-defined only for a classically conformal theory and mainly at the one-loop level. On the contrary, the relations (13) and their higher-order generalizations [22,21] can be used far beyond the framework of (17), e.g., in the non-conformal models and at higher loops. However, the consistent use of these important results is possible only on the basis of their correct interpretation. We hope that the present communication will contribute in this direction.

Nonlocal m -type form factors
Let us return to the remarkable result of [15] given by Eq.(5) suggesting an intriguing nonlocal partner for the CC term in the gravitational action. This seems to be a very good progress compared to expression (10), where one cannot find any form factor for the ρ Λ term. Therefore, first of all, we should check whether (10) is compatible with (5) derived from the same diagrams in Fig. 1.
The diagram calculation in [5] is based on the decomposition of the polarization operator in different tensor structures and using expansion (1) to the second order in h µν , where dots stand for the lower and higher orders (in h µν ) terms. Here we used the notation 2W = C 2 − E 4 which simplifies the expansion without the loss of generality. It is easy to see that there are five distinct tensor structures. In momentum space, we have [17] T 1 = δ µν,αβ = 1 2 η µα η νβ + η µβ η να ,T 2 = η µν η αβ ,T 3 = 1 k 2 η µν k α k β + η αβ k µ k ν , After decomposing the polarization operator in these tensor structures, one should calculate the loop integrals and arrive at the expressions equivalent to (10). The calculations in [15] could be done either using the gauge fixing condition ∂ µ h µ ν = 1 2 ∂ ν h for the external metric perturbation 3 (1), or simply taking into account only the first two tensor structuresT 1 andT 2 in (25). In both cases, the two tensor structuresT 1,2 in (25) are insufficient to distinguish different terms in the expansion (24). The unique remaining criterion is that the term W (or, equivalently, C 2 ) contributes to the propagator of the traceless modē h µν = h µν − 1 4 hg µν rather than the propagation of the trace mode h = h µ µ , while the R 2 -term contributes to the propagator of the trace mode only. Both aforementioned procedures (using gauge fixing for the external field h µν or taking into account only the two tensor structures instead of five), enable one to separate the two terms in (5). This means that the calculations of [15] are almost correct, but their correct interpretation requires taking into account the remaining tensor structuresT 3,4,5 in (25). Only in this way one could avoid the mixing of the contributions to the C 2 and R 2 form factors with the ones for the CC and Einstein-Hilbert terms.
Thus, the problem is to extract (5) from the full expression (10) under the gauge fixing condition ∂ µ h µ ν = 1 2 ∂ ν h, where we meet only the two tensor structuresT 1 andT 2 . Let us present the corresponding procedure in detail.
For both nonlocal form factors k W (a) and k R (a), logarithm terms are present only in the UV, thus, we have to consider the regime k 2 ≫ m 2 . In this case, by using the relations we obtain where dots indicate O(m 6 /k 6 ) and non-logarithm terms. One can extract the UV limit of the one-loop contributions (10), present it in the coordinate representation, and use the Ricci basis, i.e., replace (1/2)C 2 → W = R 2 λσ − (1/3)R 2 . This procedure gives us the following expression for the effective Lagrangian (we change the signature to the Lorentz one to have correspondence with [15]): Here the superscript (U V,2) serves to remember that the nonlocal part of this formula is the UV piece, up to the second order in m 2 /k 2 , of the full expression (10).
The first observation about expression (28) is that leading O(m 0 /k 0 ) logarithm terms in the coefficients correspond to the divergences of the fourth-derivative terms in (10), as it has to be. The sub-leading logarithmic terms have contributions O(m 2 /k 2 ) and O(m 4 /k 4 ). The last kind of terms is of special interest to us in order to reproduce (2). The non-local terms in (28) come from the fourth-derivative form factors in the general expression (10) and are not connected with the renormalization of the CC term.
The reduction of the nonlocal O(m 4 /k 4 )-terms from expression (28) to formula (5), which we reproduced from Ref. [15], requires one more operation. In the arguments of the logarithms in (27) and (28), one has to perform an ad hoc change, replacing k 2 → k 2 +m 2 and → +m 2 , respectively. In principle, this is a legitimate operation because the whole expression (28) is an expansion valid only in the UV regime m 2 ≪ k 2 and making such a replacement does not modify the coefficient of the logarithm terms. 4 After the described operation, the second order, O(m 4 /k 4 ), terms in the brackets of (28) produce the desired structures m 4 / 2 ln 1+ /m 2 exactly like in (5). It is worth noting that ln 1 + /m 2 is different from the correct nonlocal form factor given by Eq. (12). There is a qualitative similarity between the two expressions, but it is not quantitative. For instance, using the simplified formula ln 1 + /m 2 , one cannot reproduce the correct coefficient in the Appelquist and Carazzone decoupling theorem in QED [10], as this requires the correct expression (12).
Finally, to complete the comparison with formula (5), let us consider the minimal theory with ξ = 0 or, equivalently, withξ = −1/6. In this way, preserving only the O(m 4 /k 4 ) logarithmic terms, we arrive at an analog of (5), extracted from the full form factor (10), Here and in other similar formulas presented below, superscript (U V,2,nl) means that this is the expression for the second order in m 2 /k 2 nonlocal terms, extracted from the UV limit of the fourth derivative terms in the complete formula (10). The apparent difference between the two expressions (up to the replacement → +m 2 discussed above) (5) and (29) is only in the value of coefficients, which we cannot explain with a complete certainty. Let us note that the the ratio between R 2 λσ and R 2 terms in (5) may be caused by the aforementioned ambiguity (gauge fixing vs ignoringT 3,4,5 ) that is inevitable when taking into account only the two tensor structuresT 1 andT 2 in (25). This ambiguity does not exist if one takes into account all five structures in (25) and do not restrict the external metric h µν by a gauge fixing condition. This is the way calculation of diagrams was done in [5] and independently checked by the heat kernel method and in the subsequent works [9]. For this reason, we believe that the formula (28) and (as a particular case) formula (29), derived as the sub-leading expansions from the correct UV result, are correct.
The main point is that contributions (29) do not correspond to the genuine Einstein-Hilbert R-term or to the CC term. Clearly, they are some pieces of the form factors k W and k R of the fourth-derivative terms. This conclusion is proved by the analysis of the tensor structure of the polarization operator [5]. Let us stress again that this analysis is possible only taking into account all five tensor structures in (25). It looks impossible to extract some information about the nonlocal contributions to the CC term from the logarithmic part of the fourth-derivative form factors, especially because those are leading and sub-leading UV contributions, while the CC term manifests itself in the IR.
It is instructive to provide results similar to (29) in the cases of massive fermion and vector fields. The expressions similar to (28) can be easily derived from the form factors given in [6], but we do not include these technical details here. For the fermion field, we find It is interesting that, compared to the scalar field, in the R 2 λσ term there is the change of sign typical for the divergences of the m 4 -type (however, the coefficient is −2 instead of −4). On the other hand, the R 2 term cancels in this case. Thus, the similarity with the m 4 -type divergences in (10) is a pure coincidence.
Finally, for the massive vector (Proca) model, we have L (U V,2,nl) P roca = 3m 4 4(4π) 2 1 R λσ ln m 2 Up to the overall factor, this expression coincides with that for massive scalar field (29). We leave to the interested reader to check that this coincidence concerns only the O(m 4 ) nonlocal term and does not take place in the zero-and second order terms of the expansion analogous to (28). Thus, physical interpretation of the O(m 2 /k 2 ) and O(m 4 /k 4 ) terms is subtle. In the UV regime (where m 2 in arguments of logarithms becomes irrelevant), these terms really have a global scaling property identical to that of the CC term. Even if the tensor structure of these terms forces us attributing them to the C 2 and R 2 sectors of the action, the scaling properties are those of the lower-derivative terms. In this respect, one may use these expansions as an alternative to the anomaly-induced actions for the quantum massive fields at high energy (see, e.g., [29] and further references therein).
It is known that the anomaly-induced action modified for nonzero masses of quantum fields has interesting cosmological applications [29,30]. It would be certainly very interesting (albeit certainly more difficult) to explore the effects of masses by using the complete nonlocal action (10). It is worth noting that in the recent work [31] one can find a useful formalism for the massless logarithm form factors. One may expect that this interesting method can be adapted for a theory with nonzero masses. In this case, terms (2) and (30) will appear in the action as the second order corrections, while the first order terms (which we derived but not analysed here) are the leading corrections.

Conclusions
We analysed non-locality of quantum corrections due to free massive scalar, fermion, vector fields to the effective gravitational action. Although the genuine cosmological term does not admit a non-local form factor, the form factors in the C 2 and R 2 terms produce contributions (29) which at energies larger than the corresponding particle mass look like an effective cosmological term. However, there is an important difference, as these non-local analogs of the cosmological constant are, in fact, the second-order terms in the m 2 /k 2 expansions of the fourth derivative terms in the UV. Certainly, it would be very interesting to determine the impact of these and other similar low-mass terms on the evolution of the early Universe. It is worth adding that the role of f (1/ R) terms in nonlocal cosmology was analysed in [32,33]. In any case, the O(m 4 )-terms in the expression (5) do not mean the running of the cosmological constant, but represent the subleading terms in the momentum-subtraction running of coupling constants of marginal operators R 2 and C 2 .
Similarly, calculations performed in a de Sitter background should be correctly interpreted as those which describe the running of the fourth derivative (Gauss-Bonnet and R 2 ) terms, rather than the scale-dependence of the cosmological term. It is interesting that these terms do not produce higher derivative ghosts on a flat background and play very important role in the Starobinsky inflationary model [34]. Thus, the de Sitter -based relations such as (13) may be a useful alternative to the anomaly-induced effective action, especially for the phenomenologically interesting large values of the nonminimal parameter ξ.