Effective approach to the Antoniadis-Mottola model: quantum decoupling of the higher derivative terms

We explore the decoupling of massive ghost mode in the $4D$ (four-dimensional) theory of the conformal factor of the metric. The model was introduced by Antoniadis and Mottola in [1] and can be regarded as a close analog of the fourth-derivative quantum gravity. The analysis of the derived one-loop nonlocal form factors includes their asymptotic behavior in the UV and IR limits. In the UV (high energy) domain, our results reproduce the Minimal Subtraction scheme-based beta functions of [1]. In the IR (i.e., at low energies), the diagrams with massive ghost internal lines collapse into tadpole-type graphs without nonlocal contributions and become irrelevant. On the other hand, those structures that contribute to the running of parameters of the action and survive in the IR, are well-correlated with the divergent part (or the leading in UV contributions to the form factors), coming from the effective low-energy theory of the conformal factor. This effective theory describes only the light propagating mode. Finally, we discuss whether these results may shed light on the possible running of the cosmological constant at low energies.


Introduction
The running of the cosmological constant at low energies represents an interesting alternative to the numerous models of Dark Energy, as it provides the equation of state which is close, but not identical to the ω Λ = −1, of the cosmological constant.On the other hand, there is no full understanding of whether such a running is possible or not, such that this issue remains uncertain and is a subject of phenomenological considerations, as discussed in [2,3] and many subsequent works.The main difficulty for the thorough theoretical investigation is that the traditional approach to quantum decoupling [4] implies calculating the nonlocal form factor (or its equivalent) and taking its low-energy limit.The cosmological constant acquires physical sense only in curved spacetime and, in principle, the corresponding form factors have to be built from covariant elements and analysed in curved space.According to the Appelquist-Carazzone theorem [4], heavy degrees of freedom decouple in the IR regime, and their loop corrections are quadratically suppressed.The same effect should hold in curved spacetime, leading to the corresponding decoupling theorems.
The described program has been fulfilled in a series of papers [2,[5][6][7] where the nonlocal form factors in the vacuum (gravitational) actions were calculated and analysed.The problem is that, these nonlocal form factors describe the decoupling, but only for the fourth-derivative terms in the action.Owing to covariance, the form factors depend on the d'Alembertian operator .The positive powers of this operator give zero when acting on the cosmological constant and produce surface terms when acting on the scalar curvature R. Let us note that part of the mentioned papers, Refs.[6,7], include the discussion of the form factors of surface terms (see [8] for the latest discussions of the mathematical aspects of the problem), and there may be even interesting applications of the running of Newton constant, related to these surface terms.However, it is unclear how one can gain information about the running of the cosmological constant in the traditional covariant framework.
The situation changes dramatically if we perform a conformal transformation.For instance, using the parametrization with the traceless hµν and constant scale parameter M , transforms the cosmological constant term √ −g into φ 4 plus φ 4 hn -vertices.It is known that there is no problem to find nonlocal form factor and verify IR decoupling for the φ 4 -term in the scalar theory [9] and one should expect this to be equally easy in the gravitational version of the theory.Unfortunately, the described approach does not constitute a comprehensive solution of the problem of the running cosmological constant.In particular, it is not obvious that such a non-covariant running will preserve the structure of the φ 4 hn -vertices, such that the running can be safely attributed to the cosmological constant and not to the artificial scheme of reparametrization.Anyway, the running of the cosmological term in the 4D (four-dimensional) theory of the conformal factor of the metric is an attractive object of study, starting from the first proposal [10] and its realization by Antoniadis and Mottola [1].The model of quantum conformal factor follows the idea to perform secondary quantization of the anomaly induced effective action of vacuum.This action appears as a result of integrating conformal anomaly [11,12] coming from the quantum effects of matter fields (see, e.g., [13] for the review or [14] for the textbook level introduction).The simplest realization of the anomaly induced action is a theory of a single scalar field with fourth derivatives, on a flat background.This procedure corresponds "switching off" the hµν -mode in the parametrization (1).
In the paper [1] it was shown that such a model, with additional Einstein-Hilbert and cosmological terms, is renormalizable and, in particular, describes the running of the cosmological term.The remaining question is whether this running holds in the low energy domain or only in the UV.Indeed, this is a general question that is quite relevant for all higher derivative models of quantum gravity.These models may be renormalizable [15], or even superrenormalizable [16] and this enables one to consistently derive the renormalization group equations for the effective charges.In the 4D case, the beta functions are partially ambiguous [17][18][19], while in the six-or higher-derivative models, all beta functions do not depend on the gauge fixing conditions [20].However, in which physical situations the corresponding running can be applied?The one-loop corrections behind the beta functions come from the three different types of diagrams: (i) with internal lines of the massless degrees of freedom (gravitons); (ii) with internal lines of massive components, i.e., higher derivative ghosts (or ghost-like states, ghost tachyons, etc) and normal degrees of freedom, typical for the superrenormalizable models; (iii) with mixed (massless and massive) internal lines.The standard approach to effective quantum gravity [21] assumes that only the first and third types of diagrams give relevant contributions in the IR and that these contributions are the same as in the effective model where the propagators and vertices are constructed from the action of GR.As a consequence, the IR limit of an arbitrary model of quantum gravity corresponds to the quantum GR, which does not have massive degrees of freedom.As with all reasonable assumptions, this statement has to be verified.A relevant question, posed in [22] and, in a more explicit form, in [23], was whether it is possible that, instead of quantum GR, the IR theory of quantum gravity may be based on some other theory, e.g., based on a nonlocal action, such that there are still no massive degrees of freedom propagating in the IR.Answering this question requires making one-loop calculation in the momentum-dependent scheme of renormalization, in a theory with higher derivatives.Such calculation is very complicated and it looks reasonable to consider a toy model which possesses the same main features (i.e., massive and massless particle contents owing to higher derivatives and non-polynomial interactions).This kind of a model would enable one to perform necessary calculation in a more economic way.
It is easy to note that the theory of quantum conformal factor [1] represents a nearly perfect toy model for the fourth-derivative quantum gravity.The Lagrangian of this theory includes non-polynomial interactions in the two-derivative and zero-derivative sectors, similar to the fourth derivative quantum gravity.This means, the general structure of the relevant diagrams includes all the aforementioned (i), (ii) and (iii)-types.Regardless the calculation of the form factors in the momentum subtraction scheme in the theory [1] are rather involved (as the reader may see in what follows), they are still alleviated compared to the ones in a full version of quantum gravity, where one has to face more extensive set of degrees of freedom and complicated tensor structures, typical for diagrammatic treatment of quantum gravity.
In the present work, we report on the derivation of nonlocal form factors in the fourthderivative model of quantum conformal factor and perform the analysis of the UV and IR asymptotic behaviour of these quantum corrections.It is important to note that the effective approach to the theory of conformal factor induced by anomaly has an independent interest.In the recent paper [24], it was shown that this theory provides, in the effective approach, a propagation of a scalar mode of the gravitational field, which is not present in GR.In our opinion, the investigation of quantum IR decoupling is useful for a better general understanding of this model in the effective framework.
The paper is organized as follows: In sect.2, we briefly review the four-derivative model for the conformal factor and present the derivation of its UV divergences using the heatkernel method.The corresponding expression will be used, in what follows, as a reference to verify the main result in the UV.In sect.3, we formulate the elements of Feynman technique, i.e., the propagator and vertices for the model, and consider the diagrams producing ultraviolet (UV) divergences.Furthermore, we derive the one-loop corrections, including the nonlocal form factors in the propagator sector.Section 4 includes a description of the asymptotic behavior of nonlocal contributions to the two-point function in the UV and IR limits.In sect.5 we discuss the connection between the momentum dependence in the IR regime of the fundamental theory and the divergences in the effective low-energy model containing only the light (massless) mode.As usual in massless theories, the divergences define not only UV, but also the IR behaviour of the theory and can be used for comparison with the IR limit of the full theory.In Sec. 6 we present a discussion of the implications of the IR decoupling for the cosmological constant problems.Finally, in Sect.7, we draw our conclusions and discuss the possibilities of a subsequent work.The four Appendices complement the main text.In the Appendix A, one can find the set of the Feynman diagrams used in our calculations, while in Appendix B, we collect intermediate formulas concerning the calculation of Feynman integrals in dimensional regularization.Contributions to the two-point function from divergence-free diagrams are shown in Appendix C, and in Appendix D we present the complete expressions of the one-loop quantum corrections to the three-and four-point vertices.
The notations include the Minkowski signature (+, −, −, −).Also, to reduce the size of the formulas, we avoid indicating the +i in the denominators of the propagators.Indeed, the loop calculations were performed in Euclidean signature.

The model
Let us start with a brief review of the model which we shall use in the present work.The action of the model is the simplest form of the solution of the anomaly-induced action [11,12], with the flat fiducial metric, plus the Einstein-Hilbert and cosmological terms, Here κ = 8πG and the coefficients w, b and c are the one-loop semiclassical beta functions in the vacuum sector, where N s , N f , N v are the multiplicities of the quantum conformal matter fields of spins zero, one-half and one, respectively.The trace anomaly which produces the induced part of the action (2) is The coefficient c can be modified by adding a finite local term R 2 to the action S anom (see [13,[25][26][27] for detailed discussion).This feature will not affect our considerations, especially because we will not need particular versions of the beta functions (3) and concentrate on the general features of the quantum theory of conformal factor based on (2).On top of induced part, the action includes Einstein-Hilbert and cosmological terms, which are not renormalized at the initial semiclassical theory, but become very relevant at the second stage, when we quantize the conformal factor.
The idea that the conformal factor can be quantum, despite it emerges as an effective mode in the integration of matter fields, comes from Polyakov's approach in 2D, related to string theory [28].The idea of using the equivalent metric-scalar (Liouville) model as the basis of 2D quantum gravity was quite popular in 90-s.The use of the analogous theory (in curved spacetime) as a model for 4D quantum gravity was proposed in [10].In four dimensions, the theory for the conformal factor can be regarded as a truncated version of the four-derivative quantum gravity at large distances (i.e., for the low energies, or IR), providing a screening mechanism for the cosmological constant [1].An important difference with the 2D induced gravity is that, in 4D one can add the classical terms.Alternatively, one can make the Einstein-Hilbert and cosmological terms to be generated in the scheme of induced gravity [29], but this requires an independent scalar field and does not fit our purpose to construct a simplified model to explore the decoupling in a higher derivative quantum gravity.
As any fourth-derivative quantum gravity model, the model of our interest has massive modes, which can be ghosts and tachyonic ghosts. 1 The question of our interest is what happens with the contributions of these massive modes at low energies.
It proves useful to introduce notations similar to [1], such that the action (2) becomes The difference in notations with the paper by Antoniadis and Mottola is the coefficient of the kinetic sector of higher derivative terms θ 2 , which is denoted Q 2 /(4π) 2 in [1].On top of this, we fix the anomalous scaling dimension α = 1 to avoid additional complications of formulas.
The last two terms in (6) come from the Einstein-Hilbert and cosmological constant terms.In the IR, these terms dominate over the higher derivative terms and it proves useful to split the Lagrangian into two terms, i.e., and Our plan is to evaluate the quantum corrections in full theory (6) and, separately, for the theory based on the IR-term (7).Due to the presence of higher derivative terms, the one-loop divergences in the full theory are obtained using the generalized Schwinger-DeWitt technique [17,31].
Using the background field method, the conformal factor is decomposed into classical σ and quantum ρ counterparts, σ → σ + ρ.Then we obtain the bilinear in the quantum field forms for the two terms, and The Hermitian forms for the structures ( 9) and ( 10) are obtained as So, for the complete model ( 6), we have where the self-adjoint four-derivative minimal operator is with the elements Using the standard algorithm for the fourth-order operators [17,31], we arrive at the expression for the divergences where we introduce the useful notation ε = (4π) 2 (n − 4) and neglect the irrelevant surface terms.This result agrees with the previous calculations [1,32], except for an apparent misprint in the sign of Eq. ( 4) of [32].An additional verification can be found in the recent paper [33].
In that follows, we shall confirm the expression (15) by the calculation of both divergent and finite nonlocal (leading logarithms) parts of the Feynman diagrams.By considering the minimal subtraction (MS) scheme, one can easily derive UV β-functions for the theory (6).In the next section, we will determine the finite parts of the one-loop diagrams that produce these divergences.In this case, the structure (15) will be useful in identifying the diagrams that are relevant for our purposes.For example, from the coefficients of the terms (∂σ) 2 e 2σ and e 4σ , we can expect that diagrams involving interaction vertices γ and γζ provide contributions to the Einstein-Hilbert sector, while diagrams with vertices λ and γ 2 are associated with corrections to the cosmological constant sector.Later on, we shall see that this identification may not hold in the effective low-energy model with two derivatives.
For completeness, we also derived the divergences of the effective theory, based on the IR-term, Eq. ( 7), separately.The result is As it should be expected from the power counting, the fourth-derivative counterterms are required in this theory, as it is non-renormalizable.At the same time, neglecting the fourthderivative terms according to the effective approach, we arrive at the reference expression to compare with the IR limit of the full theory.

One-loop corrections from Feynman diagrams
In a model with higher derivatives, to explore the decoupling in the loop corrections, one has to separate massive and massless degrees of freedom.In many cases, this can be achieved by introducing auxiliary fields (see, e.g., [34]).However, in the case of the theory (6), this approach is not operational owing to our interest in the quantum corrections in the theory that have higher derivatives in both kinetic terms and the interactions.Thus, we shall make the separation at the level of the propagator and vertices in the Feynman diagrams, i.e., use the method close to the one of [35].
The structure of the vertices and the propagator for the fundamental theory (6) can be calculated by using the parametrization σ → σ + ρ, where ρ is a small perturbation and expanding the exponential terms in the power series in ρ.Collecting the quadratic terms, we find that the propagator satisfies the equation Making the Fourier transform, and assuming Λ γ 2 /θ 2 , we get Finally, in the same approximation, the propagator can be written as It is easy to identify a healthy degree of freedom with the mass m 2 = 8Λ/3 and a ghostly mode with the Planck-scale mass, M 2 = γ/θ 2 .We need to consider only those interaction vertices that are relevant for the one-loop corrections to the propagator.The vertices for the 3-and 4-point functions arise from the derivative interaction terms in the part L 4der and from the higher order terms in the exponential expansion in L IR , In Fig. 1, we have presented the vertices corresponding to these interaction terms.The analytic expressions have the form and where (k • p) = k µ p µ .Now we are in the position to determine the one-loop contributions to the self-energy (correction to the propagator) of the healthy (light) mode.First, consider the diagrams producing the UV divergences (15), derived previously using the heat-kernel method.These divergences are responsible for the MS-scheme based beta functions and serve as the UV references for the complete expressions.
In the theory (6) the expression for the two-point function is where the omitted (. ..) terms denote contributions from the divergence-free diagrams.On top of this, the finite contributions Σ λζ , Σ γλ and Σ λ 2 are proportional to the mass m 2 (as shown in Appendix C).As we shall see in Sec. 4, these terms can be neglected in our analysis of the IR limit, when we consider m 2 = 0. We can write the general symbolic expression for the self-energy function, in the secondorder (in coupling constants) approximation as presented in Fig. 2, for corrections of the type Σ, and in Fig. 3, for corrections Σ. order (in coupling constants) approximation as presented in Fig. ??. and where solid lines indicate light degrees of freedom while dashed lines stand for the massive ghosts.Furthermore, order (in coupling constants) approximation as presented in Fig. ??.
and Now we are in the position to determine the one-loop contributions to the propagator of the healthy (or light) mode.For this we consider the self-energy diagrams that produce the UV divergences (16).In the theory (6) the corrections of interest for the two-point function are given by The self-energy type corrections that are second-order in the coupling constants are given by and where, generically, where solid lines indicate light degrees of freedom while dashed lines stand for the massive ghosts.Furthermore, 10 Furthermore, in Figs. 2 and 3, the expressions for the light, ghost and mixed sectors are, respectively, (25) while the contributions of the first-order in the coupling constants have the general structure shown in Fig. 4, where solid lines indicate light degrees of freedom while dashed lines stand for the massive ghosts.Furthermore, while the contributions of the first-order in the coupling constants have the following structure: The diagrammatic representation of the contributions to the two-point function with di↵erent couplings, is presented in the Appendix A. The diagrams in Fig. 2 correspond 10 Figure 4: "Snail" diagrams associated with the corrections Σ.In this case, obviously, there are no contributions from a mixed sector.
The diagrammatic representation of the contributions to the two-point function with different couplings, is presented in the Appendix A. The diagrams in Fig. 5 correspond to the fifth term in (24), and the last term is associated with the diagrams in Fig. 7.In Fig. 6 there are shown the diagrams that correspond to the term proportional to γζ.The diagrams for the first-order terms in the couplings γ and λ, are depicted in Figs. 8 and  9, respectively.In addition, the third and fourth terms in (24) are associated with the tadpole diagrams with interaction vertices γλ shown in Fig. 10, and λ 2 shown in Fig. 11.
Let us note that each diagram here represents the sum over all the topologically equivalent diagrams with different permutations over the external momenta and with all possible placements of derivatives on the internal and external lines.On top of that, we omitted some tadpole-type diagrams that do not contribute to G (2) (p, −p), as they include derivatives of the propagator in a single spacetime point, and hence vanish.
To evaluate the integrals in (24) we used dimensional regularization.In the model under consideration, this requires extending the standard list of divergent expressions [36] for the integrals in the spacetime of 2ω complex dimensions.The integrals proportional to ζ2 , γζ and γ 2 read, respectively, as and where we used the following combinations of the vertex factors: The results of the integrations in the Euclidean space are 2 where and γ E ≈ 0.577 is the Euler-Mascheroni constant.In the limit ω → 2, the results ( 31), ( 32) and ( 33) represent divergent and finite parts.Note that the finite part of these expressions has a very complicated dependence on the external momentum.For these nonlocal structures, in the mixed sector, we used the notation Furthermore, the notations used in the light and ghost sectors, include, respectively, It is worth explaining how the definitions ( 35) and ( 36) can be used in identifying contributions coming from the different sectors.E.g., the logarithmic form factors involving c, in the results presented above, come from the "pure" loops with only the propagators of the light degrees of freedom (equivalently, for the other momentum-dependent logarithmic structures).Of course, there are terms that result from the combination of contributions from the three sectors (i.e., light, massive ghost and mixed), such as ln (1 + 4b) and those with only polynomial dependencies on the external momentum.In top of this, the relevant corrections involving the quartic vertices, are given by where the integral is The results (37) are tadpole-type contributions, which do not produce a nonlocal form factor.Therefore, these corrections are not relevant to our analysis at low energies and were included just of completeness.Of course, the same consideration applies to secondorder tadpole-type corrections, which in principle contribute to divergences in the Einstein-Hilbert and cosmological constant sectors, with m 2 M 2 = 8λ/θ 2 .Actually, the corrections in (39) can be disregarded because tadpole diagrams, such as those presented in Figs. 10 and 11, normally are eliminated using renormalization conditions (see, e.g., [39] or the Chapter 11 of [40] for more details).
Let us note that here we presented only the results for the self-energy.The lower-order vertices were also derived and produce qualitatively the same picture.The nonlocal parts of these contributions follow a standard logarithmic structure, as those for propagator corrections.Since the corresponding formulas are relatively bulky, they are separated in Appendix D.

Asymptotic behavior
In this section, we explore the asymptotic behavior of the one-loop contributions ( 31), (32), and (33).Our main interest is to verify how these expressions interpolate between the UV and IR regions of the fundamental theory.In this way, we have a chance to understand what happens to the nonlocal form factors of the contribution of loops with the massive degrees of freedom (massive ghosts) in the IR.
We start with the limit p 2 → ∞ that corresponds to the UV regime p 2 M 2 m 2 .In this case, Eqs. ( 32), (31), and (33) simplify and we arrive at the expressions As expected in the UV regime, the leading logarithmic terms in the form factor, i.e., the terms with ln p 2 /µ 2 , are proportional to the corresponding divergences.It is easy to verify that, when returning to the coordinate representation, the divergent part of the expressions above, together with (37) in the UV, correspond to the result (15), obtained from the heat-kernel technique.
On the other hand, assuming m 2 = 0 in the formulas of Σ ζ 2 , Σ γζ and Σ γ 2 (see part B.2 of Appendix B), the analysis of the IR regime M 2 p 2 of these corrections provides The last formulas show that, in the IR limit, the divergences and momentum dependence do not correlate with each other, exactly as it is expected [4] (see also [2,5,7] for the semiclassical theory).We have found that this basic feature holds also for the "mixed" diagrams, such that the Appelquist-Carazzone theorem is valid for the fourth-derivative model with non-polynomial interactions.In the expressions ( 43) and ( 44), the nonlocal part with logarithmic form factor ln p 2 /M 2 is suppressed by powers of M 2 , whereas in (45) this is not the case, as the factor γ 2 (remember that γ = θ 2 M 2 ) cancels this suppression in the terms proportional to p 4 .Although it may not be obvious, one can check that these nonlocal structures represent the contributions from the light sector alone.The one-loop diagrams with mixed (light and massive ghost) internal lines and (of course) the pure ghost contributions collapse and produce only trivial dependencies on the external momentum.
All in all, we verified the quadratic decoupling of the heavy mode in the Feynman diagrams with the mixed contents.

One-loop corrections in the effective theory
The last element of our investigation is the comparison between what remains from the logarithmic form factors of the full theory in the IR and the leading logarithms in the effective (initially local) theory without heavy degrees of freedom.According to the existing expectations [21], the two expression should demonstrate a perfect correlation.This result would mean, in particular, that the quantum general relativity can serve as a universal low-energy model in any renormalizable or superrenormalizable approach to quantum gravity.
So, let us evaluate the quantum corrections to the propagator in the effective low-energy model of ( 6), containing only the light mode.We consider a scenario in which the energy scale is much smaller than the Planck mass.Therefore, we can assume that the EH and cosmological constant terms dominate over the higher derivative terms, leaving only the part L IR .Under these considerations, the tree-level propagator of the conformal factor boils down to where m 2 is defined in (20).The vertices are the same as those in ( 22) and ( 23).Since we are dealing with an effective model, we are not concerned that L IR is non-renormalizable, as we may ignore the higher-derivative divergences3 .Thus, our interest is to explore the contributions to the cosmological constant and the Einstein-Hilbert terms.These formulas can be compared with the structures found in the IR limit of the full theory (6).
In the low-energy effective theory, the relevant contribution is given by To make the comparison more explicit, consider the particular case Λ = 0 (or, equivalently, Then the last expression reduces to a simpler form, Note that the nonlocal contribution ln p 2 /M 2 involving the interaction vertex γ 2 in the IR regime of the "fundamental" theory (6), Eq. ( 45), correlates with the logarithmic term of the result (48), regardless in the IR of the fundamental theory there is no UV divergence.From these results, it is possible to establish the one-loop match between these two scenarios, i.e., fundamental and effective.Let us note that the emergence of p 4 factor in the effective approach and its identification with the cosmological constant term has been discussed in the recent literature.In particular, this issue was described in detail, exactly as a reaction to the attribution of the part of the gravitational form factor to the cosmological constant in [43].It reality, the corresponding terms appear as part of the expansion of the nonlocal form factors of the R µν R µν and R 2 terms and have no direct relation to the cosmological constant term [42].
In order to establish the match between the fundamental and effective scenarios, we introduce the relation where δ eff γ 2 is an additional term which represents, at the one-loop level, the difference between the correction of the fundamental theory in the low energies and the correction of the effective theory.Considering the collapse of the diagrams with massive ghost internal lines in the IR regime of the fundamental theory, as we saw in the previous section, we can identify that the additional term in ( 49) is composed of contributions arising from the collapse of loops in the mixed sector.These collapsed diagrams reduce to the tadpole-type graphs, and the remaining part, related to pure ghost loops, i.e., The contribution of the tadpole-type in ( 50) is proportional to the mass m 2 , and hence vanishes in the simplification adopted for the IR, namely assuming c (1) γ 2 , mixed m 2 =0 = 0. Using the results ( 45) and (48) in relation (49), we find the leading logarithmic terms in the form In the above expression we present only the finite part, since the divergences can be removed by a suitable renormalization procedure.As it should be expected, the nonlocal part with momentum-dependent logarithmic form factor is canceled and the IR matching condition, which ensure the equivalence with the result (45), is satisfied with δ eff γ 2 contains only terms with trivial dependencies on the external momentum.

Implications for the cosmological constant problem
The cosmological constant problem is one of the main unsolved issues in the presentday theoretical physics.The problem was formulated by Weinberg in [44] as the need to explain the extremely precise fine tuning between the original cosmological constant density in the vacuum action and the huge induced contributions.One can reformulate the problem in terms of the renormalization of the vacuum term [45] but this does not help too much in its resolution.There are also many other interesting aspects of the problem, related to cosmology (see, e.g., [46,47]).Along with the main problem, in the quantum field theory framework we need to understand whether the cosmological constant density and the Newton constant are really constants or these parameters can be slowly varying with the energy scale, as predicted, e.g., by the four-derivative model of [1] (see also the examples of discussions based on the extended models [32,48,49] and supersymmetric generalization [50]) and, of course, in the full fourth-derivative [17,18,22] or even higherderivative models [20] of quantum gravity.
As we saw in the previous sections, the naive Minimal Subtraction -based approach to the renormalization group for the cosmological constant term is not operational, as it ignores the decoupling of the massive (ghost or healthy, in some models) degrees of freedom.Assuming that all massive degrees of freedom have typical masses of the Planck order of magnitude, all the cosmological applications occur at the deep IR, where the Appelquist-Carazzone -type decoupling changes the beta functions.The question is what remains from these beta functions in the theory with both massive and massless degrees of freedom [22,23]?
The result which we got for the quantum theory of conformal factor is that, in the deep IR, there remain the contributions ( 43), ( 44) and (45), which fit the ones of effective low-energy quantum theory based on the local model.This provides a positive answer to one of the main questions posed in [23] and confirms the hypothesis of [21].We got a strong confirmation that the IR limit of a higher derivative model of quantum gravity, which has massive degrees of freedom (those may be healthy modes or ghosts) corresponds to the loop corrections in the effective quantum gravity based on general relativity.The result comes from the model of [1] which has non-polynomial interactions, very close to the case of real models of quantum gravity.In this sense, the new confirmation is more explicit than the results obtained previously in the framework of simplified models, such as [51]).Looking at the remnant expressions of the form factors in the IR, we note that the terms without p 4 or p 2 have only log(µ 2 /M 2 ) coefficients and no momentum-dependent logarithmic terms.This is consistent with the previous analysis of the possible quantum corrections, indicating that the logarithmic (in Euclidean momentum) form factors cannot be inserted in the cosmological constant term [2].Recently, similar observations were done, e.g., in [24,52,53]).This situation confirms the general expectation that there cannot be physical running of the cosmological constant term, detectable by means of flat-space calculations, as discussed in [2] and more recently in [42].Let us stress that this does not mean that the cosmological constant running is impossible in general, it just cannot be detected in the flat-space calculations [3].
It is worth noting that one can observe the running of the cosmological constant term using the non-covariant parametrization such as (1), just as a decoupling in the beta function of the φ 4 -interaction.Up to a certain extent, the corresponding calculations were already developed in [9] and can be generalized to other theories, including quantum gravity.This would be certainly an interesting way to extend the present work.However, it is important to be careful with the expectations to the results of such an extension, as there will always remain a question about the physical interpretation of the result obtained by means of non-covariant methods.

Conclusions and discussions
We have considered the detailed renormalization in the theory of the conformal factor.Assuming a small cosmological constant, such a theory possesses two mass scales with a strong hierarchy between them.On top of that, the theory has non-polynomial interactions and is renormalizable [1].These features make the model qualitatively similar to the higher derivative quantum gravity.Previously, the one-loop calculations in this model were performed in the Minimal Subtraction scheme and we performed a more detailed analysis in the momentum subtraction scheme of renormalization.
The analysis of the nonlocal form factors shows that in the UV, we meet a correspondence with the Minimal Subtraction scheme results.On the other hand, in the IR we met a strong deviation from this simplified scheme and, as it was anticipated, a good agreement with the calculation in the effective model that ignores the massive degree of freedom.One of the new details is that the "mixed" diagrams, with the internal lines of both smallmass and large-mass fields, transform into tadpoles.These diagrams contribute to the UV divergences, but not to the nonlocal form factors.This means, the diagrams with the large-mass internal lines collapse and become irrelevant in the IR.In particular, nonlocal structures that survive in this regime and contribute to the cosmological constant sector, are correlated with the UV divergent part in an effective version of the model containing only the light mode.
It would be certainly interesting to extend the analysis which was presented above, for the models of "real" quantum gravity, i.e., the theory of quantum metric.As we mentioned in the Introduction, this is a technically more challenging problem because such a theory has gauge invariance and complicated tensor structures in the sectors of quantum metric and ghost.However, the results presented above show that there are very good chances to meet the expectation of universality of quantum general relativity as an effective theory of quantum gravity, at least in the fourth derivative [15] and, probably, all polynomial models introduced in [16], where all extra degrees of freedom have the masses of the Planck order of magnitude [54].
At the same time, the situation may be more complicated in the non-local theories of quantum gravity [55][56][57][58] (many further references can be found in the last review).The most popular version of nonlocal models are free from massive degrees of freedom at the tree-level.On the other hand, starting from the one-loop level, the structure of the propagator changes and there are infinitely many complex-energy and complex-mass ghostlike states with the quasi-continuous mass spectrum [59].In this case, the universality of general relativity as the IR quantum gravity theory is rather uncertain.This means, there are still many interesting issues to explore in the area of the present work.
The last point is that we have found a good correspondence between the IR limit of the theory with massless and large-mass degrees of freedom and the UV limit of the effective theory without the massive particles.This correspondence extends to the contributions in the cosmological constant sector of the gravitational action.However, these contributions are not momentum-dependent, confirming the general no-go statement [2] concerning the detection of the cosmological constant running by means of the flat-space calculations.On the other hand, this output does not means that such a running is impossible by itself.Instead, it should be interpreted as a challenge to develop new methods of effective field theory calculations which would be appropriate for clarifying this issue.

Appendices A Feynman diagrams
In this appendix, we present the set of one-loop Feynman diagrams that correspond to the corrections to the two-point function.
1 Appendix -Feynman diagrams 1 1 Appendix -Feynman diagrams In this appendix, we present the set of one-loop Feynman diagrams that correspond to the corrections for the two-point function.(2)          ⌃

B Intermediate results
In this appendix, we present some intermediate results related to the calculation of Feynman integrals in sect.3.

B.1 Feynman integrals
By using the Feynman parametrization and performing the following shift of integration variable k = q + px, we can rewrite the integrals related to the mixed sector in the expressions ( 27), ( 28) and ( 29), respectively, as and where, in Minkowski space, Here we define ∆ ≡ p 2 x(x − 1) + (M 2 − m 2 )x + m 2 .For the integrals in the other sectors, we have the same results as above with ∆ = ∆ ghost = p 2 x(x − 1) + M 2 in the case of the ghost sector, and ∆ = ∆ light = p 2 x(x − 1) + m 2 for the light sector.
B.2 Corrections Σ ζ 2 , Σ γζ and Σ γ 2 for the case of m 2 = 0 The contributions from the diagrams shown in Figs. 5, 6 and 7, assuming m 2 = 0, can be written, respectively, as ζ (p) ln ζ, light (p) ln ζ, ghost (p) ln ζ, mixed (p) ln Σ γζ (p) γζ (p) ln γζ, light (p) ln and γ, light (p) ln γ, mixed (p) ln where β's are coefficients of the nonlocal part with momentum-dependent logarithmic form factor, decomposed according to the light, ghost and mixed sectors, while α's and ξ's are coefficients of contributions involving the combination of different sectors: γ, light (p) = γζ, ghost (p) = − p 2 (8M 4 − 2M 2 p 2 − p 4 ) 2M 4 d , β ζ, mixed (p) = − ζ (p) = p 4 5 − γζ (p) = p 2 3 + We collect here the results of the self-energy corrections that are free of divergences.The contributions from the diagrams shown in Figs.In this appendix we present the set of Feynman diagrams associated with corrections for the propagator and vertices.Here we assumed the sum over the diagrams of the same topology, but with di↵erent permutations of the external momenta.

Appendix -Feynman diagrams
In this appendix we present the set of Feynman diagrams associated with corrections for the propagator and vertices.Here we assumed the sum over the diagrams of the same topology, but with di↵erent permutations of the external momenta.The analytic expressions corresponding to these diagrams are Γ (4) γζ (p, r, q) where, for the s-channel diagrams, Γ (4,4) (76)

Figure 1 :
Figure 1: Feynman diagrams associated with the interaction vertices.The primes denotes derivatives acting on the propagators.

Figure 2 :
Figure 2: General structure of one-loop diagrams for self-energy functions Σ. Solid lines indicate light degrees of freedom while dashed lines stand for the massive ghosts.

G ( 2 )⇣ 2 Figure 1 :
Figure 1: Diagrams for the two-point function that provide one-loop contributions to the renormalization of the coupling ⇣.Solid lines indicate light degrees of freedom, dashed lines stand for the massive ghosts, and the primes denotes derivatives acting on the propagators.* E-mail address: wagnorion@gmail.com

Figure 5 :
Figure 5: Diagrams for the two-point function that provide one-loop contributions to the renormalization of the coupling ζ.Solid lines indicate light degrees of freedom, dashed lines stand for the massive ghosts, and the primes denotes derivatives acting on the propagators.

Figure 2 :
Figure 2: Diagrams for the two-point function with the interaction vertex ⇣.

Figure 6 :
Figure 6: Diagrams for the two-point function with the interaction vertex γζ.

Figure 2 :Figure 3 :Figure 4 :
Figure 2: Diagrams for the two-point function with the interaction vertex ⇣.

G ( 2 )Figure 2 :
Figure 2: Divergence-free diagrams for the two-point function with the vertex .* E-mail address: wagnorion@gmail.com

Figure 12 :
Figure 12: Divergence-free diagrams for the two-point function with the vertex λζ.

Figure 1 :
Figure 1: Divergence-free diagrams for the two-point function with the vertex ⇣.

G ( 2 )Figure 2 :
Figure 2: Divergence-free diagrams for the two-point function with the vertex .* E-mail address: wagnorion@gmail.com

Figure 13 :Figure 3 :
Figure 13: Divergence-free diagrams for the two-point function with the vertex λγ.

Figure 14 :
Figure 14: Divergence-free diagrams for two-point function with the vertex λ 2 .

Figure 1 :
Figure 1: Diagrams for the three-point function with the interaction vertex ⇣ 2 .

Figure 2 :
Figure 2: Diagrams for the four-point function with the interaction vertex ⇣ 2 .* E-mail address: wagnorion@gmail.com

Figure 15 : 2 (Figure 3 :
Figure 15: Diagrams for the three-point function with the interaction vertex ζ 2

Figure 4 :
Figure 4: Diagrams for the four-point functions with the interaction vertex ⇣.

Figure 16 :
Figure 16: Diagrams for the three-point function with the interaction vertex γζ.

Figure 5 :Figure 17 :
Figure 5: Diagrams for the three-point function with the interaction vertex 2 .

Figure 1 :
Figure 1: Diagrams for the three-point function with the interaction vertex ⇣ 2 .

Figure 2 :
Figure 2: Diagrams for the four-point function with the interaction vertex ⇣ 2 .* E-mail address: wagnorion@gmail.com

Figure 18 :
Figure 18: Diagrams for the four-point function with the interaction vertex ζ 2 .

Figure 3 :
Figure 3: Diagrams for the three-point functions with the interaction vertex ⇣.

Figure 4 :
Figure 4: Diagrams for the four-point functions with the interaction vertex ⇣.Figure 19: Diagrams for the four-point function with the interaction vertex γζ.

Figure 19 :
Figure 4: Diagrams for the four-point functions with the interaction vertex ⇣.Figure 19: Diagrams for the four-point function with the interaction vertex γζ.

Figure 5 :
Figure 5: Diagrams for the three-point function with the interaction vertex 2 .

Figure 6 : 3 Figure 20 :
Figure 6: Diagrams for the four-point function with the interaction vertex 2 .