The Conformal Anomaly Action to Fourth Order (4T) in $d=4$ in Momentum Space

We elaborate on the structure of the conformal anomaly effective action up to 4-th order, in an expansion in the gravitational fluctuations $(h)$ of the background metric, in the flat spacetime limit. For this purpose we discuss the renormalization of 4-point functions containing insertions of stress-energy tensors (4T), in conformal field theories in four spacetime dimensions with the goal of identifying the structure of the anomaly action. We focus on a separation of the correlator into its transverse/traceless and longitudinal components, applied to the trace and conservation Ward identities (WI) in momentum space. These are sufficient to identify, from their hierarchical structure, the anomaly contribution, without the need to proceed with a complete determination of all of its independent form factors. Renormalization induces sequential bilinear graviton-scalar mixings on single, double and multiple trace terms, corresponding to $R\square^{-1}$ interactions of the scalar curvature, with intermediate virtual massless exchanges. These dilaton-like terms couple to the conformal anomaly, as for the chiral anomalous WIs. We show that at 4T level a new traceless component appears after renormalization. We comment on future extensions of this result to more general backgrounds, with possible applications to non local cosmologies.


Introduction
nonvanishing trace of the stress energy tensor and defines a link with dark energy and the cosmological constant in gravity.

Trace anomalies
In any CFT a crucial role is played by the stress energy tensor T µν and by its trace, both in the definition of the CFT and in the description of its breaking, due to the presence of a conformal anomaly in even spacetime dimensions [43]. As shown in several previous analysis, such a breaking is characterised by the appearance of massless poles [36,[44][45][46][47] in specific form factors of correlation functions involving one or more stress energy tensors, in the form of bilinear mixings. Such contributions are ubiquitous in all the (chiral, conformal, superconformal) correlation functions investigated so far, indicating that their appearance is directly linked to an explicitly broken phase induced by the anomaly, due to renormalization. Bilinear mixings are, even in ordinary field theory, the signature that a functional expansion is taking place in a nontrivial vacuum, as for the Higgs mechanism. In an ordinary gauge theory such mixings are removed by a suitable gauge choice, such as the 't Hooft or the unitary gauge. In this case, the massless pole amounts to a virtual (nonlocal) interaction, which is directly coupled to the anomaly. As we are going to elaborate in a following section, the bilinear mixings correspond to an expansion of the same action -in coordinate space and in the flat limit-in the dimensionless variable R −1 . Here, R is the scalar curvature, which has very often appeared in the analysis of nonlocal cosmologies, such as in f (R −1 ) models. In this case, the goal has been that of explaining the late-time dark energy dominance of our universe. These analysis have been limited in the past to 3-point functions. Four-point functions have received attention only more recently [48][49][50][51]. For 3-point functions, the analysis of conformal and non-conformal correlators in realistic theories such as QED and QCD, at one-loop, has covered the T JJ and the T T T [36][37][38]46]. Although such bilinear nonlocal interactions have been identified in free field theory, it has been shown that they are equally present in the general explicit solutions of the CWIs [52], once a matching between such solutions -which for 3-point functions are uniquely determined modulo few constants - [41] and the perturbative realization [39], is found. Such correspondence provides a considerable simplification of the general solution, which can then be expressed just in terms of simple one-loop integrals [37,38]. It has been shown in some detail, in the TJJ case at least, that the emergence of such interpolating states can be matched with the non-perturbative solutions (i.e. the form factors) identified from the transverse traceless sector of such correlator [52]. In this work we are going to make a decisive step forward, by showing that this feature is generic and is completely controlled by the renormalized anomalous CWIs satisfied by the correlators. No complete solution of the CWIs at quartic level (O(h 4 ) in flat space, with h the metric fluctuation) is needed in order to extract such behaviour, at least in flat space, once the contribution of the anomaly is correctly taken into account. Notice that the advantage of solving the CWIs, as in the case of conformal 3-point functions, beside its undisputable value, is that it gives the opportunity to apply and verify the renormalization procedure on the solution, using the known gravitational counterterms, as given in Eq. (2.59) below. However, for the rest, the method is increasingly prohibitive if the goal is to infer the all-order behaviour of the anomaly action. The purpose of our paper is to show that a significant amount of information is hidden in the structure of the CWI's, if we assume that they can be renormalized using the standard (known) counterterms typical of the anomaly functional. We work our way from this assumption backward, in order to characterize the structure of the CWIs and identify the implications for the anomaly action to a given order in h µν (h), the fluctuation of the metric around its flat spacetime limit. Our approach is, in a way, quite direct, and can be extended to n-point functions and to any spacetime dimension, following the same strategy. More details on these further developments will be presented in forthcoming work. Here we intend to characterize the method in its simplest formulation, by working in d = 4 spacetime dimensions, relying on an expansion of the anomaly functional around a flat background in Dimensional Regularization (DR).
We believe that these analysis are necessary in order to nail down the structure of the anomaly action using a direct approach, independently of those entertained so far, based on the variational solutions of the anomaly equation, which take either to a nonlocal Riegert action [53] or, alternatively, to local ones, such as those derived by the Noether (leaking) method (see for instance [54,55]), with the inclusion of an asymptotic dilaton. Variational solutions of the anomaly equations are naturally limited by the mathematical procedure of integration of the underlying action, since they differ by arbitrary Weyl-invariant contributions. This has raised several issues concerning the consistency of these results.
Our work is organised as follows. We will define our conventions and characterise the main features of the anomaly action in section 2, highlighting all the simplifications that are present when we consider the flat spacetime limit in the definition of the correlation functions. Conveniently, the discussion, in this section, is carried out in full generality, and provides a wider perspective on the structure of the effective action (Γ) for arbitrary metrics, and on the increased complexity of the CWIs around such backgrounds. It is well known that in more general backgrounds, for instance for Weyl flat/conformally flat metrics, the structure of the quantum average of the energy momentum tensor (its one-point function), is affected by tadpole contributions which are fixed by the anomaly, and need to be included in the analyis, and call for a generalization of our method. Then we will review the structure of the CWIs in the longitudinal/transverse separation introduced in [41] and developed further in [40,42,56], followed by the analysis of the anomalous CWI's. In section (4.2) a new derivation, performed directly in d = 4, of the special CWI's is presented for the 4T. The approach extends the one developed in [57] for the 3T to the new case. The renormalization of the 4T and its longitudinal/transverse decomposition is worked out in a follow up section, where, just for completeness, we also classify the singular form factors which are affected by renormalization. The structure of the anomaly action is slowly built starting from the 2-point function, that we review. The vanishing of the anomaly contribution to the anomaly action Γ at O(h 2 ) is illustrated in great detail. Then we derive the structure of the local (i.e. t loc or longitudinal stress energy tensor) contributions to the CWI in the renormalization procedure, and separate the anomaly contributions from the unknown but finite, renormalized parts. This is the content of section (7.2). Even if the finite local terms are not explicitly given, we show that they are not necessary in order to identify the structure of the anomaly action at the level of the 4T. The result is obtained by working directly with the longitudinal components t loc of the counterterms to the same 4T correlator, combined with their trace WIs. We show that a structure with bilinear mixings appears quite naturally from the decompositon, together with an extra, trace-free contribution. This extra term, absent in the 3-point function, appears for the first time at the level of the 4T. A comment on our result follows next, before our Conclusions.

The anomaly action
In general, the effective action, Γ[g] (for former discussions of Γ[g] see [58][59][60]) which is a functional of the external metric background, can be generated, for instance, by integrating out some conformal matter in the path integral, leaving the external metric arbitrary. The simplest realization is provided, for instance, by a scalar free field theory, discussed in several perturbative studies [37,38,51]. Compared to Sakharov's induced gravity, where integration over ordinary matter, for a generic metric g, is expected to generate terms of the form [61] Γ ef f ∼ d 4 x √ g Λ + c 1 (g)R + c 2 "R 2 " , (2.1) corresponding to a cosmogical constant, the Einstein-Hilbert action and to generic "R 2 " terms, the integration over conformal matter should not introduce any scale, if the result if the integration turns out to be finite. This turns out to be the case, in d spacetime dimensions, at least for a scalar theory, as far as we stay away from even dimensions, but a renormalization procedure is required in the limit d → 4, with the inclusion of a renormalization scale µ. This appears as a balancing factor µ − -with = d − 4 -in the structure of the counterterms, which are expressed in terms of Weyl-invariant operators in d = 4. The renormalized action then acquires a log k 2 /µ 2 dependence, where k is a generic momentum, but such terms are part of its Weyl-invariant part, while its anomaly part is simply associated to a welldefined pole structure, on which we are going to elaborate below. This is associated to R −1 dimensionless operators attached to the external legs of the correlator. We will come back to comment on this result in a final section. For a general CFT, when no Lagrangian is present, the constraints induced by the CWI's are identical to those derived from the functional integral (Lagrangian) approach. In this case Γ[g] can be simply defined to be the functional which collects all the correlation functions with single and multiple T-insertions, not necessarily related to a path integral formulation. Therefore, the CWIs remain identical both for Lagrangian and non-Lagrangian CFTs. In the latter case one could resort to the operatorial content of the theory and, for primary scalar fields, derive the CWIs in a completely independent manner (see for instance [62,63] and [38] for coordinate and momentum space derivations of tensorial correlators). We recall that in an ordinary field theory the relation between the partition function and the functional of all the connected correlators is obviously given by the functional relation e −Γ(g) = Z(g) ↔ Γ(g) = − log Z(g). (2.2) As mentioned, Z(g) can be thought of as related to a functional integral in which we integrate the action of a generic CFT over a field (φ), or in general, a collection of fields, in a background metric g µν Z(g) = N Dφe −S(g,φ) , Z(η) = 1. Its logarithm, Γ(g), is our definition of the effective action. As usual Z[g], in the Feynman diagrammatic expansion, will contain both connected and disconnected graphs, while Γ[g] collects only connected graphs. It is also easy to verify that this collection corresponds also to 1PI (1 particle irreducible) correlators, since we are not allowing external scalar sources (J φ ) nor gravitational fluctuations in the quantum corrections. The emergence of bilinear mixing on the external graviton lines, as we are going to realize at the end of our analysis, should be interpreted as a dynamical response of the theory, induced by the process of renormalization, with the generation of a dynamical degree of freedom. For this reason, the emergence of such terms does not invalidate the 1PI nature of this functional. As a reference for our discussion, as already mentioned, we may assume that S(φ, g) describes, for instance, a free scalar field φ in a generic background. The action, in this case, is where we have included a conformal coupling , and R is the scalar curvature. This choice of χ(d) guarantees the conformal invariance of this action in d dimensions as well as introduces a term of improvement for the stress-energy tensor in the flat limit, which becomes symmetric and traceless. A general perturbative analysis of this term and its role in the renormalization of the stress energy tensor of the Standard Model can be found in [64,65]. Dimensional counting on this action can be performed, as usual, in two ways, either by considering the canonical dimension of the fields -as determined by their kinetic term in the Lagrangian -or by their response to Weyl transformations. In the first case we have (in mass-scale dimensions) Flipping the sign of σ obviously changes the counting from mass-scale to length-scale dimensions. The action associated to (2.4) is obviously scale invariant. The dimensional counting for Weyl scaling proceeds differently. The mass scaling of a length scale ds 2 (ds 2 → Ω 2 ds 2 , Ω = e −σ ) is accounted for only by the metric g µν → e 2σ g µν .
(2.6) while keeping the same scale dimensions of the scalar field, and one obtains , in this case, collects correlation functions with multiple insertions of the stress energy tensor Anomalous dimensions appear as soon as we switch-on an interaction in (2.4). For instance, a λφ 4 /4! potential will induce anomalous dimensions and a β function which in flat space (d = 4 − 2 ) take the form and induce a breaking of the classical conformal symmetry. For this reason we stick to the λ = 0 case. Γ[g] can be computed in a generic background, but its structure and the conditions that it has to satisfy vary considerably with g.
If we consider a generic metric background, there are significant changes on the structure of the CWIs. The flat spacetime limit corresponds to the simplest nontrivial case. Results obtained in this case, using the gravitational formulation of Γ[g], are equivalent to those obtained for ordinary conformal field theory in flat space, which can be naturally defined without any reference to gravity. It is also clear, from this correspondence, that the renormalization of such theories, in such limit, should involve only counterterms of mass dimension four.
In the most general case, one can derive CWI's in backgrounds of the form (g µν ), with g µν = g µν + δg µν (δg µν = h µν ), and the formalism that we are going to discuss needs to be extended in order to encompass also this scenario. In general, the variation (2.6) on the functional Γ[g] takes to the relation and its invariance under Weyl δ σ g µν = 2σg µν (2.11) and diffeomorphisms δ g µν = −∇ µ ν − ∇ ν µ , (2.12) summarised by the relations δ σ Γ = 0 δ Γ = 0, (2.13) take to trace and conservation conditions of the quantum averages of T µν (2.14) Trace and conformal WI's can be derived from the equations above by functional differentiations with respect to the metric background. The relations above are modified in tthe presence of an anomaly. The anomalous trace WIs can be derived by allowing for an anomaly contribution on the rhs of the σ variation in (2.13) where √ gĀ(x) is the anomaly. Functional differentiations of this relation take to the hierarchy of trace WIs that we will be defining below. The identification of the special CWIs in the formalism of the effective action, in the presence of an anomaly, is less straightforward that in flat space, and can be addressed in this formulation starting from the definition of the conformal current and its quantum average. This is defined in the form is a Killing vector field of the metric g. The proof follows closely the classical geometric derivation of the conservation of such current. We recall that Then, the requirement of diffeomorphism and Weyl invariance, summarised by the two conditions (2.14), take to the conservation of the conformal current Notice that if Γ satisfies (2.14), it is sufficient to require that the metric allows KVs, for the corresponding conformal current J c in (2.16) to be conserved. If we require that the metric background allows conformal Killing vectors (CKVs) and the effective action is invariant under such transformations, then we have the possibility to include the anomaly contribution in the equations. We recall that the CKVs are solution of the equation Notice that if we introduce a conformal current, now using the CKVs of the background metric as in (2.16) Notice that this relation can still be imposed as a conservation equation on correlators of the form J c T in d dimensions, since the anomaly contribution is induced only at d = 4 while the stress energy tensor is always conserved. This is the approach described in section Section 4.2 that we will exploit in the derivation of the special CWIs. Notice that σ(x) is, at the beginning, a generic scalar function, which in a Taylor expansion around a given point x µ is characterized by an infinite and arbitrary number of constants. Their number gets reduced drastically if we require that the spacetime manifold with metric g allows a tangent space at each of its points, endowed with a flat conformal symmetry. Indeed, in flat space, the conformal Killing equation takes to (Killing) vectors µ at most quadratic in x, parameterised by the 15 parameters (a µ , ω µν , λ s , b µ ) of the conformal group, indicated as K µ The analysis is done in d=4, following the approach of [57], which takes directly to anomalous special CWIs. In our conventions, n-point correlation functions will be defined as with √ g 1 ≡ |det g µ 1 ν 1 (x 1 ) and so on.
2) collects all the connected contributions of the correlation functions in the expansion respect to the metric fluctuations, and may as well be expressed in a covariant expansion as Diagrammatically, for a scalar theory in a flat background, it takes the form where the external weavy lines represent gravitational fluctuations, in terms of contributions that are classified as tadpoles, 2-point, 3-point and n-point correlation functions of stress energy tensors. Tadpoles are removed in DR in flat space, and the sum in (2.25) starts from 2-point functions. The anomaly contributions start from 3-point functions. Expression (2.25) needs to be renormalized by suitable counterterms. In a free-field theory realization the expansion covers only one-loop diagrams and the origin of the anomaly lies entirely in the renormalization of such contributions. The structure of the counterterms is crucial in order to identify its renormalized expression.

The expansion in a general background
The correlation functions collected into Γ(g) are obviously affected by contact terms, which can be clearly identified. These stem from the definition of the nT's correlators -due to the inclusion of the √ g/2 prefactor in (2.8) -but also from the quantum averages of multiple functional derivatives of the stress energy tensor, that we are going to classify. We generalize the example discussed above, by turning to the expansion of a generic functional of the metric, still denoted as Γ[g], now for a generic CFT. Γ[g], as already mentioned, collects all the nT's multi-point functions, which are hierarchically constrained by the symmetry of the corresponding CFT. The expansion, similarly to the free-field theory example in (2.25), also in this case can be conveniently thought as generated by the integration of a conformal theory characterised by a fundamental action S.
In order to simplify the index structure of the equations, we will introduce some notations. For instance we define δ n Z(g) In the case of the 4-point correlator, which will be relevant for our discussion, we have a similar expansion, which in terms of the averages of the classical action and its derivatives takes the form  (2.29) and so on. They can be re-expressed in terms of correlators of lower orders (2-and 3-point functions) and of contact interactions. Diagrammatically, the expansion of the 4T correlator of Γ takes the form where the averages correspond to direct insertions of δS/δg operators for each separate external graviton vertex. Notice that the "S" correlators, which contain insertions of single and multiple derivatives of the fundamental action, do not correspond, according to the definition (2.23), to insertions of stress energy tensors, but differ from the latter by a certain number of contact terms, that we intend to identify. It is convenient, for this purpose, to introduce some condensed notation, in order to avoid, at least in part, the proliferation of indices in the analysis below. For instance, the relation will be summarized below in the expression where dg(a) ≡ g(x a ) symbolized the metric determinant at point x a , (a = 1, 2, 3, 4) and δ(a, b) ≡ δ(x a − x b ). Differentiation is performed, in our conventions, with respect to the metric with covariant indices (g µaνa ) on the left-hand side of the equations, but the result on the right-hand side has all the indices contravariant, e.g. g(a) ≡ g µaνa , T (a) ≡ T µaνa . Contact terms are generated either by differentiation of the metric, as above, or by multiple differentiation of the action S. Using the notation above, the stress energy tensor T is related to S in the form δS δg(a) an differentating respect to the metric this relation we obtain the identity where the differentiation of the stress energy tensor δT(b) δg(a) ∼ c(g)δ(a, b) introduces one contact term. Contact terms of higher orders are present in the other derivatives, for example and Contact terms generated by higher derivatives of the fundamental action can be found in Appendix A.
Correlators with multiple T 's and those with S-insertions are then related by contact terms involving expectation values of a single T and of their derivatives. For instance, for 2-point functions, the relation between the S and T correlators, the latter defined by (2.23), takes the form where we have used the tadpole relation (2.34). The expression above shows that the 2-point function of stress energy tensors -defined as T T -and the result obtained by direct S-insertions, agree if we are allowed to remove all the tadpoles from the expansion of the functional. Since we will be defining the hierarchy of the CWI's in flat space, this is possible if we adopt a regularization scheme in which the tadpoles vanish. This requirement, in the context of a conformal theory, has a deep physical meaning and is regularization dependent, for being associated to a diverging quartic contribution, if the computation is performed covariantly with the inclusion of a UV cutoff (Λ 4 ). Such cutoff dependence epitomizes the hierarchy problem for the cosmological constant, which is clearly hidden in DR, since massless tadpoles, in this regularization scheme, are set to zero. A recent analysis [66] has concluded that this issue, even if the computation is performed covariantly and by a cutoff, can be ameliorated by the inclusion of extra singular interactions coming from the path integral measure, not considered before. Since our analysis is performed in flat space we will proceed by eliminating such contributions, by adopting a DR scheme.
There is a large list of such contributions which are removed from our analysis as g → δ, for a flat background. In this case we obtain where δ a ≡ δ µaνa , obtained by the flat limit g(a) → δ a and in a general background these contact terms would be related to additional conformal data, not present in our analysis. Notice that (2.40), in a diagrammatic expansion, defines a connected functional if we resort to free field theory and allow the ordinary Wick contractions of the corresponding operators on its rhs. Indeed, in the case of the scalar free field theory presented above, the topological contributions coming from the 4-T are just summarized by the vertices which can be directly computed in perturbation theory [51]. If we indicate by . . . c the connected contributions to this expansion, then the expression of the 4T in the flat limit, using DR, takes the form A similar result holds for the 3-point function T T T , from (2.27) Notice that (2.45) and (2.46) are derived from a path integral realization of a certain CFT, in the flat spacetime limit. The condition of integrating out over the conformal matter in the path integral forces us to consider only one-loop terms generated by the ordinary Wick contractions, accompanied by an arbitrary number of external graviton lines. This is equivalent to going from a general expansion (2.30) to the simplified one (2.44). It is also clear that this result is specific of the flat background case. For instance, in the conformally flat (Weyl-flat) case where g µν = e 2σ(x) η µν (2.47) this expansion requires the inclusion of massless tadpoles. In particular, the expansion of the dominant contribution to Γ would start with a single insertion of stress energy tensor contracted with the gravitational fluctuation ∼ T · h , and would be entirely defined by the anomaly functional. The tadpoles would contribute to all orders in an expansion h. This is related to the fact that the entire stress energy tensor is proportional to the anomaly coefficient of the Gauss-Bonnet term, as discussed in [67,68].

Polynomiality of the counterterms
If we expand around a flat metric background and rely on a mass independent regularization scheme, then the structure of the counterterms is simply polynomial in momentum space. Indeed, in general, the breaking of Weyl invariance takes to the anomalous variation which is constrained by the Wess-Zumino consistency condition to take the form given in terms of the dimension-4 curvature invariants which are the Euler-Gauss-Bonnet (GB) invariant and the square of the Weyl conformal tensor, respectively, in d = 4.
We pause for few comments. From now on, when referring to C 2 without any subscript, we will be indicating the (C (d) ) 2 expression of this invariant, with a parametric dependence on d (2.54) The choice of (C (d) ) 2 instead of (C (4) ) 2 in the counterterm action that we will define below, takes to variations which are deprived of total derivative ( R) term in (2.50). A R term is indeed generated by a finite renormalization, by varying a (d − 4)R 2 term. With this choice of C 2 one derives the relation which differs from an analogous one obtained by the replacement of C 2 → (C (4) ) 2 in the integrand, by a finite renormalization. The number of such invariants depends on the dimension. The case of d = 6 is discussed in [69]. One can derive an anomaly action of the Wess-Zumino form starting from these invariants using the Weyl gauging approach [70,71], with the case of d = 6 worked out in [55]. The procedure, however, in that example, is quite different from the one followed in this work, in which the anomaly action is just identified by the analysis of the CWIs after renormalization. Indeed, the approach of Weyl gauging introduces one extra degree of freedom in the spectrum (a dilaton), and identifies just another variational solution of the anomaly constraint. The result is a local action, with extra interactions in the dilaton field which depend on the spacetime dimensions. All of this is avoided in out approach, since we retain into Γ[g] only the genuine constraints coming from the anomalous CWI's, with no inclusion of any extra compensator field.
In generic (even) spacetime dimensions, the structure of the countertem Lagrangian, is modified accordingly, with the Euler/ GB density, given by which, for d = 4, is quadratic in the curvatures, and is indeed given by (2.51). It is the latter, together with the other invariant C 2 , the only extra one present in d = 4, to be varied around d = 4 in order to generate the counterterms, and satisfies the relation In the expressions above, all the traces are performed in d spacetime dimensions, as in ordinary DR, and the renormalization of the entire functional is then obtained by the addition of the counterterm action corresponding to the Weyl tensor squared and the Euler density in d = 4, as in (2.51) and (2.52), and µ is a renormalization scale, while ε = d − 4. Notice that the scaling dimensions of the counterterms are fixed to their d = 4 value, and only the parametric dependence on d -for (C (d) ) 2 -is modified as we analyitically continue d around d = 4. For a generic nT correlator, the only counterterm needed for its renormalization, is obtained by the inclusion of a classical gravitational vertex generated by the differentiation of (2.59) n times. The renormalized effective action Γ R is then defined by the sum of the two terms The renormalized correlator shown above satisfies anomalous CWIs.

The inclusion of the anomaly
To characterize the anomaly contribution to each correlation function, we start from the 1-point function.
In a generic background g, the 1-point function is decomposed as being the trace anomaly equation, and T µν f being the Weyl-invariant (traceless) term. Following the discussion in (2.50), these scaling violations may be written for the 1-point function in the form -having dropped the suffix Ren from the renormalized stress energy tensor -where the finite terms on the right hand side of this equation denote the anomaly contribution with being the anomaly functional. In general, one also finds additional dimension-4 local invariants L i , if there are couplings to other background fields, as for instance in the QED and QCD cases, with coefficients related to the β functions of the corresponding gauge couplings. For n-point functions the trace anomaly, as well as all the other CWIs, are far more involved, and take a hierarchical structure. For all the other WIs, in DR the structure of the equations can be analyzed in two different frameworks.
In one of them, we are allowed to investigate the correlators directly in d spacetime dimensions, deriving ordinary (anomaly-free) CWI's, which are then modified by the inclusion of the 4-dimensional counterterm as d → 4. In this limit, the conformal constraints become anomalous and the hierarchical equations are modified by the presence of extra terms which are anomaly-related. Alternatively, it is possible to circumvent this limiting procedure by working out the equations directly in d=4, with the inclusion of the contributions coming from the anomaly functional, as we are going to show. This second approach has been formulated in [57] and will be extended to the 4-point function in Section 4.2.
We recall that the counterterm vertex for the nT correlator, in DR, takes the form Figure 1 Expansion of the anomaly contributions to the renormalized vertex for the TTT. and are the expressions of the two contributions present in (2.67) in momentum space. One can also verify the following trace relations Obviously, the effective action that results from the renormalization can be clearly separated in terms of two contributions, as clear from (2.63), corresponding to an anomaly part Γ A [g] and to a finite, Weyl-invariant term, which can be expanded in terms of fluctuations over a backgroundḡ as for the entire effective action Γ[g] (2.73) This functional collects finite correlators (2.62) in d = 4. A similar expansion holds also for Γ A , the anomaly part.
The anomaly effective action Γ A that results from this analysis in momentum space is a rational function of the external momenta, characterised by well-defined tensor structures, and it is free of logarithmic terms, as shown in direct perturbative studies of the TJJ and TTT [36,45,46]. In the case of the T T T all the contributions to the trace anomaly of the renormalized vertex can be summarised by the three diagrams shown in Fig. 1 and will be discussed below. In this figure, the dashed lines indicate the inclusion of an operator on the external lines, which projects on the subspace transverse to a given momentum p. The projector is accompanied by a pole. Together,π and the pole define the ordinary transverse projector inserted on all the external lines in one, two and three copies. Poles in separate variables 1/ , are connected to separate external graviton lines, and each momentum invariant appears as a single pole. This result had been obtained in [37] by performing a complete perturbative analysis of the same vertex using free field theory realizations. In this case, the inclusion of 3 sectors, a scalar, a fermion and a spin 1, allows to obtain the most general expression of this correlator, and its renormalization, performed by the addition of the general counterterm (2.59), has been verified. Our goal in the next few sections will be to show that the Weyl-variant contribution to Γ A can be identified directly from the CWIs, under the assumption that Γ ct , as defined in (2.59), is all that is needed in order to proceed with the renormalization of a 3-or a 4-point function of correlators of stress energy tensors in d = 4, in the flat spacetime limit. If we move to coordinate space from momentum space and attach the metric fluctuations h µν on the external lines, it is easy to show that the inclusion of a projector of the form (2.75) induces a bilinear mixing between the graviton and a virtual scalar propagating in the intermediate virtual (dashed) line. The interaction takes the form In the T T T case, as one can immediately figure out from Fig. 1, such expressions can be easily traced back to nonlocal terms in the anomaly action Γ A of the form where the labels (1), (2) refer to the first and second variations of the invariants R, the scalar curvature, C 2 and E 4 . Analogously, in the case of the TJJ correlator in QED, the anomaly contribution, extracted by a complete perturbative analysis, is shown in Fig. 2 and takes the form This characterization of the anomaly action Γ A at cubic order, in each of the two cases, has been obtained by a complete analysis of the conformal constraints, using the explicit expression of their solutions, which for 3 point functions depends only on three constant for the TTT and on two for the TJJ. In both cases, the solutions, which are given by hypergeometric functions F 4 , can be mapped to general free field theory realizations, characterized by the inclusion at 1-loop level of an arbitrary number of scalars, fermions and spin 1 fields in the Feynman expansion. The map between the two approaches is an exact one in d = 4, 5.
In [57] it has been shown that such nonzero trace contributions are automatically generated by a nonlocal conformal anomaly action of Riegert type [72] expanded to third and to first order in δg, respectively. As we move from 3-to 4-point functions, the solutions of the CWIs are affected by the appearance of arbitrary functions, which is a general tract of CFT, and this approach is not of immediate help, in the sense that even if such bilinear mixings would be found in the perturbative description of a certan correlator, they would not find an immediate counterpart in the general solution of the conformal constraints.
Obviously, we would like to provide a proof of such behaviour with no reference to free field theory. We recall that the CWIs' impose, at any n, hierarchical relations connecting n to n − 1 point functions, which we are going to investigate in great detail in momentum space. While these relations are not sufficient, for a generic CFT, to reconstruct the expression of Γ[g] beyond O(δg 3 ) (i.e. n = 3), they nevertheless constrain significantly its structure after renormalization.
As we are going to show, the CWIs predict a well-defined structure for the anomaly contribution to Γ A at the level of the 4T, once we assume that their renormalization proceeds via the counterterms action (2.59). As already mentioned, this analysis does not require a complete classification of all the form factors which appear in the correlator, but just a careful study of the structure of the CWIs satisfied by it and by the TTT. For this reason, we are going to illustrate how this simplified procedure works first in the case of the 3T, before moving to the 4T in this new approach, showing how the structures of Γ A is constrained by the renormalized CWIs to assume a form very similar to the 3T case. As in previous studies in momentum space, we will rely on decompositions of the 3T and 4T into a transverse traceless, a longitudinal, and a trace part, extending the approach formulated in [41] for 3-point functions to 4-point functions.

Conservation Ward identities
The anomaly action Γ A [g] is constrained by a hierarchical set of equations which can be derived by the symmetries of the general effective action Γ. We proceed assuming that the correlation functions can be derived by varying the path integral definition of Γ[g] (2.2) (2.3) as in (2.23). Starting from the covariant definition of the stress-energy tensor, expressed in terms of a fundamental action as in (2.8), but for the rest, generic this 1-point function satisfies the fundamental Ward identity of covariant conservation in an arbitrary background as a consequence of the invariance of Γ[g] under diffeomorphisms. Here (g) ∇ µ denotes the covariant derivative in the general background metric g µν (x). It can be expressed in the form where Γ µ λν is the Christoffel connection for the general background metric g µν(x) . Our definitions and conventions are summarised in an Appendix. In order to derive the conservation WIs for higher point correlation functions, one has to consider additional variations with respect to the metric of (3.2) and then move to flat space, obtaining are the first and second functional derivatives of the connection, in the flat limit. We have explicitly indicated the symmetrization with respect to the relevant indices. We have defined δ , and introduced a simplified notation for the Dirac delta δ x i x j ≡ δ(x i − x j ). All the derivative (e.g. ∂ λ ) are taken with respect to the coordinate x 1 (e.g.∂/∂x λ 1 ). We Fourier transform to momentum space with the convention Here we have used the translational invariance of the correlator in flat space, which allows to use momentum conservation to express one of the momenta (in our convention p 4 ) as combination of the remaining onesp 4 = −p 1 − p 2 − p 3 . Details on the elimination of one of the momenta in the derivation of the CWIs and on the modification of the Leibnitz rule in the differentiation of such correlators in momentum space be found in [38]. The conservation Ward Identity (3.4) in flat spacetime may be Fourier transformed, giving the CWIs in momentum space where we have defined related to the second and first functional derivative of the Christoffel connection respectively.

Conservation WI's for the counterterms
To illustrate the conservaton WI in detail, we turn to the expression of the counterterm action (2.59), which generates counterterm vertices of the form where on the rhs of the expression above we have introduced the counterterm vertices (with P = 12) evaluated in the flat spacetime limit. These vertices share some properties when contracted with flat metric tensors and the external momenta. In particular, one can verify the validity of the following relations in d dimensions which play a key role in the renormalization procedure. Furthermore, the contraction of these vertices with the external momenta generates conservation Ward Identities similar to (3.8), where C and B are given in (3.9) and (3.10). These equations can be generalized to the case of n-point functions.

Conformal Ward Identities
Turning to the ordinary (i.e. non anomalous) trace and conformal WIs, these can be obtained directly in flat space using the expression of the dilatation and special conformal transformations operators. The dilatation WI's for the T T T T can be easily constructed from the condition of Weyl invariance of the effective action Γ, or equivalently, in the ordinary operatorial approach (see [38]) where we have used the explicit expression of the scaling dimension of the stress energy tensor ∆ T = d.
Analogously, the special conformal WIs, corresponding to special conformal transformations, can be derived in the operatorial approach, applied to an ordinary CFT in flat space, relying on the change of T µν under a special conformal transformation, with a generic parameter b µ , and σ = −2b · x The action of the special conformal operator K κ on T in its finite form is obtained differentiating respect to the parameter b κ and using the Leibniz rule for the variation on correlation functions of multiple T's, can be distributed to the entire correlator as which takes the form where κ is now a free Lorentz index. Notice that the use of an operatorial approach, one needs to rely on correlators defined via direct insertions of T's. Such correlation functions, are, in general, different from the definition given above in (2.23) due to possible contact terms and of nonvanishing tadpoles, not present in (4.4). For this reason the CWI's derived by this method and by the functional method that we will present below, are naive expressions which are perfectly well-defined and equivalent, only in the presence of a suitable regularization scheme and of a flat background. In DR, which is well-defined in a flat spacetime, the vanishing of the 1-point function and the inclusion of vertex counterterms shows that we don't need to worry about such differences. Notice that these constraints are directly written in momentum space as and in terms of a dilatation operator D and a special conformal transformation operator K κ . The action of the differential operators on the momenta is defined implicitly on the 4th momentum, as discussed in the case of 3-point functions in previous works [38,39,41], with a modification of the Leibnitz rule.

Trace and conformal anomalous Ward identities
The CWIs become anomalous as we move from d spacetime dimensions to 4. In d dimensions the conformal symmetry of the correlator T T T T is preserved and this property is reflected in the trace identity which generates, as we have already mentioned, a hierarchy of equations by functional differentiation of this result respect to the background metric g. Equivalently, the same equations can be derived from the condition of Weyl invariance of the effective action. Following the same procedure as for the conservation WIs, we may derive the trace Ward identities for the four-point function T T T T , in general d dimensions, as which may be written in momentum space, after a Fourier transform, as We have omitted an overall δ function, having replaced p 4 withp 4 . In d = 4 spacetime dimensions the equations need to be renormalized, by adding local covariant counterterms which will be generated from the action (2.59). If general covariance is respected by this procedure, the conservation Ward Identities remain valid for the renormalized effective action and for its variations. This is reflected on the hierarchical structure of the equations, which remain identical to the bare (naive) case. Trace identities of the correlation functions involving at least three stress energy tensor operators are instead affected by the anomaly, due to the scale violations induced by the regularization/renormalization procedure. In d = 4 the corresponding anomalous Ward identities for the trace can be obtained by a functional variation of the equation (2.65) with respect to the background metric. In this case (4.9) is characterised by new contributions on its rhs, coming from the anomaly A(x) where the trace anomaly functional A is given in Eq. (2.66), and the Fourier transform of its variation in the flat spacetime limit takes the form (4.12) We will be using the general definition

The anomalous CWI's using conformal Killing vectors
The expressions of the anomalous conformal WIs can be derived in an alternative way following the formulation of [57], that here we are going to extend the 4-point function case. The derivation of such identities relies uniquely on the effective action and can be obtained as follows. We illustrate it first in the TT case, and then move to the 4T. We start from the conservation of the conformal current (2.18) (4.14) In the TT case the derivation is simplifies, since there is no anomaly with the choice of the counterterm action defined in (2.59), a point that we will address in a section below. The conservation of the conformal current J µ (K) implies the conservation relation on the integral involving the two point function J µ By making explicit J µ (x) = K ν (x) T µν (x), with → K in the flat limit, the previous relation takes the form We recall that K ν satisfies the conformal Killing equation in flat space and by using this equation (4.16) can be re-written in the form where we observe the appearance of the conservation and trace Ward identities for the two-point function T T that in the flat spacetime limit are explicitly given as The special conformal transformations are obtained by considering the form of the conformal killing vector K (C) ν , as where κ = 1, . . . , d is a d-dimensional vector index. By using (4.21) in the integral (4.18), we write and by using the conservation and trace Ward identities (4.19) and (4.20), and integrating by parts we obtain the final expression that are exactly the special conformal Ward identities for the T µ 1 ν 1 (x 1 ) .

4-point functions
The derivation above can be extended to n-point functions of the form having used the conservation of the conformal current in d dimensions under variations of the metric induced by conformal Killing vectors. In the flat spacetime limit the current, built out of these vector fields takes the ordinary form satisfying (4.17). In absence of the anomaly, the conservation of the current J µ c follows from the conservation of the stress energy tensor with the zero trace condition, as discussed in Section 2. We consider the total divergences of the form where we are assuming that the surface term vanishes, and this reflects in the behaviour of the correlation function that falls off fast enough at infinity. Expanding (4.26) and by using we find The dilatation CWI is obtained by the choice of the Conformal Killing Vector such that and the dilatation WI from (4.26) becomes and using the conservation and trace Ward identities in d = 4 for the 4-point function written as and . . . T µ 4 ν 4 (x 4 ) + (12) + (13) + (14) The anomalous Dilatation WI from (4.29) is given by where d = 4. It is worth mentioning that (4.32) is valid in any even spacetime dimension taking into account the particular structure of the trace anomaly in that spacetime. The special CWIs correspond to the d special conformal Killing vectors in flat space given in (4.21), as in the T T case. We derive the identity By using the relations (4.30) and (4.31) and performing the integration over x explicitly, we find the anomalous special CWIs for the 4-point function as where the inclusion of the anomaly, as discussed in Section 2, comes from the inclusion of the trace WI, exactly as in the T T case. At this stage, these equations can be transformed to momentum space, giving the final expressions of the CWIs in the form for the dilatation and for the special CWI's, having used the definition (4.13). When this procedure is applied to the n-point function, one finds that the anomalous conformal Ward identities are written as for the dilatation and for the special conformal Ward identities, wherep n = − n−1 i=1 p i and we have used the definition (4.13).

Decomposition of the TTTT
The analysis in momentum space allows to identify the contributions generated by the breaking of the conformal symmetry, after renormalization, in a direct manner. For this purpose we will be using the longitudinal transverse L/T decomposition of the correlator presented in [41] for 3-point functions, extending it to the 4T. This procedure has been investigated in detail for 3-point functions in [37] in the context of a perturbative approach [52]. The perturbative analysis in free field theory shows how renormalization acts on the two L/T subspaces, forcing the emergence of a trace in the longitudinal sector. Due to the constraint imposed by conformal symmetry (i.e. their CWI's), the correlation functions can be decomposed into a transverse-traceless and a semilocal part. The term semilocal refers to contributions which are obtained from the conservation and trace Ward identities. Of an external offshell graviton only its spin-2 component will component will couple to transverse-traceless part. In general, by decomposing the gravitational fluctuations into their transverse-traceless and spin-1 and spin-0 components one finds an interesting separation of the anomaly effective action which can be useful also in a phenomenological context. We will address this point in a forthcoming paper. The split of the energy momentum operator in terms of a transverse traceless (tt) part and of a longitudinal (local) part [41] is defined in the form We have introduced the transverse-traceless (Π), transverse-trace (τ ) and longitudinal (I) projectors, given respectively by with δ µν αβ = Π µν αβ + Σ µν αβ (5.10) where δ µν αβ is symmetrised with 1/2 in front. Notice that we have combined together the operators I and τ into a projector Σ which defines the local components of a given tensor T , according to (5.2), which are proportional both to a given momentum p (the longitudinal contribution) and to the trace parts. Both Π and τ are transverse by construction, while I is longitudinal and of zero trace.
The projectors induce a decomposition respect to a specific momentum p i . By using (5.1), the entire correlator is written as where the first contribution is the transverse-traceless part which satisfies by construction the conditions It is clear now that only the second term in (5.12) contributes entirely to the conservation WIs. Thus, the proper new information on the form factors of the 4-point function is entirely encoded in its transverse-traceless (tt) part, since the remaining longitudinal + trace contributions, corresponding to the local term, are related to lower point functions.

Projecting the Conformal Ward Identities
The action of D and K on the T T T T in (4.6) and (4.7), after the projection on the transverse traceless component gets simplified. We start by looking at the dilatation operator D which has the property of keeping unchanged the subspaces onto which we project the T T T T via the Π and Σ projectors. In particular we see that the action of D on the transverse traceless part of the 4-point function produces a result that is still transverse traceless. The longitudinal part of the correlator, once we apply the D operator, remains still longitudinal. This properties can be summarized as follows having used the properties of the projectors Σ and Π for which It is worth mentioning that the equations in (5.14) are trivially satisfied thanks to the linearity of D as a differential operator. This implies that the D operator does not mix the subspaces onto which we are decomposing the T T T T correlations function. For this reason, if one wants to project the equation (4.6), by using the transverse-traceless and longitudinal projectors, there are only two ways of doing it. These are the cases where we have either four Π's or four Σ's, due to the relations in (5.14). The relevant, non trivial, projected dilatation WIs are that can be simplified as once we insert the decomposition of the T T T T and use the properties (5.14). It is worth mentioning that (5.19) does not impose any additional constraints on the 4-point function. This because the longitudinal part of the correlator is explicitly given in terms of lower point functions and then (5.19) is related to the dilatation Ward Identities of the 3-and 2-point functions of stress-energy tensors. The constraints on the 4-point function will be derived from (5.18), which is related to the transverse traceless part of the correlator. Turning our attention towards the special CWIs, we observe that the action of the K κ operator on the transverse-traceless part gives a result that it is still transverse and traceless This can be shown exactly as in the case of the TTT discussed in previous works. This property of K κ allows us to identify the relevant subspace where the special CWIs act. As already pointed out, the transverse-traceless part of the T T T T is the only part of the correlator which is really unknown. Indeed, the longitudinal components can be expressed in terms of 2-and 3-point functions, by using the conservation and trace WIs. Therefore, the special CWIs will be constraining that unknown part, which can be parametrized in terms of a certain number of independent for factors, as in the 3T case. By using the properties of the projectors Σ and Π in (5.15), from (5.20), one derives the relation since the action of K κ , as just mentioned, is endomorphic on the transverse traceless sector. With this result in mind, using the projectors Π and Σ we project (4.7) into all the possible subspaces and observe that when at least one Σ is present, the equations reduce to the form The equations that result The only significant constraint will be derived when acting with 4 Π projectors. For this reason we are interested in studying the equation If we insert the decomposition (5.12) into the equation above, one can prove that terms containing two or more t loc operators will vanish when projected on the transverse traceless component, for instance where the definitions (5.5) and (5.11) have been used. Proceeding with the reconstruction program, we need the action of the special conformal transformations on the correlators with a single t loc , after projecting on the transverse traceless sector. After a lengthy but straightforward calculations we obtain are the SO(4) (Lorentz) generators and S µν is the spin part, for which S µν ρσ = δ µρ δ νσ − δ µσ δ νρ . (5.28) By using the Lorentz Ward identities we obtain the expression Analogous results are obtained for the other terms involving one t loc operator.
In summary, when we project (4.7) on the transverse traceless components we find

Identifying the divergent form factors
The general expression of the tt-contributions can be identified by imposing the transversality and trace-free (5.13) conditions on all the possible tensor structures which are allowed by the symmetries of the correlation function. The decomposition can be generically written in the form in terms of form factors A. We have explicitly labeled the form factors with an index counting the number of momenta that each of them multiplies in the decomposition. This notation will be useful in our discussion below. The choice of the independent momenta of the expansion, similarly to the case of 3-point functions, can be different for each set of uncontracted tensor indices. We will choose as basis of the expansion for each pair of indices shown above. The linear dependence of p 4 , which we will impose at a later stage, is not in contradiction with this choice, which allows to reduce the number of form factors, due to the presence of a single tt projector for each external momentum. This strategy has been introduced in [41] for 3-point functions and it allows to reduce the number of form factors. These, in eq. (5.31) are functions of the six kinematic invariants 33) or equivalently, in a completely symmetric formulation, they are functions of the six invariants As already mentioned, the local part of the 4T can be expressed entirely in terms of three-and twopoint functions due to the transverse and trace WIs. The explicit form of the local contribution is indeed given by the expression where the insertion of t loc gives Notice that the right-hand-side of (5.36) is entirely expressed in terms of lower-point correlation functions, due to the WI's (3.8) and (4.10). Similar relations hold for all the other contributions contained in (5.35).

Divergences and Renormalization
In order to investigate the implications of the CWIs on the anomaly contributions of the 4T, we turn to 4 spacetime dimensions and discuss the anomaly form of such equations. We start from the dilatation WIs. The scale invariance of the correlator is expressed through the Dilatation Ward Identity (4.6), which in terms of the corresponding form factors takes to scalar equations of the form where n p is the number of momenta multiplying the form factors in the decomposition (5.31). Eqs. (6.1) characterize the scaling behaviour of the form factors, and allow to identify quite easily those among all which will be manifestly divergent in the UV. For instance, the form factor corresponding to eight momenta in (5.31) has degree d − 8 and is finite in d = 4. This simple dimensional counting can be done for all the form factors allowed by the symmetry of the correlator. We have summarised the UV behaviour in the table below Form Factor The expected form factors that will manifest divergences in d = 4 are those of the form A (4p) , A (2p) and A (0p) in (5.31), which will show up as single poles in the regulator ε. The procedure of renormalization, obtained by the inclusion of the counterterm (2.59), will remove these divergences and will generate an anomaly. An explicit check of this cancelation is contained in [51], in the case of a conformally coupled free scalar theory.

Explicit form of the divergences
Being the anomaly generated by the renormalization procedure, it is possible to derive the structure of the anomaly contributions and the form of the anomalous CWIs' by applying the reconstruction procedure to the counterterms. One can also work out the explicit structure of the counterterms for each of the divergent form factors A (np) identified above. Their renormalization is obviously guaranteed by the general counterterm Lagrangian (2.59). For example, considering the decomposition identified in (5.31), the corresponding counterterms to the A (np) form factors can be determined in the form The scalar case, discussed in [51] can be obtained by assigning specific values to the b and b coefficients of the counterterms (2.59), expressed in terms of a scalar field content.

Renormalization and anomaly in d = 4
To illustrate the emergence of longitudinal projectors in multi-point correlation functions in d = 4, we start from the case of the T T , then move to the 3T and conclude our discussion with the 4T . The simplest context in which to discuss the renormalization of the TT is in free field theory, and include three independent sectors with n S scalars, n F fermions and n G gauge fields. A direct computation gives showing the separation of the result into a transverse-traceless (Π) and longitudinal part (π µ 1 ν 1 ). About d = 4, the projectors are expanded according to the relation where the expansion is performed on the parametric dependence of the projector Π(d). As usual in DR, the tensor indices are continued to d dimensions and contracted with a d-dimensional Euclideaan metric (δ µ µ = d). Using (7.2) in (7.1), the latter takes the form where Π (4) µ 1 ν 1 µ 2 ν 2 (p) is the transverse and traceless projector in d = 4 andB 0 (p 2 ) = 2 + log(µ 2 /p 2 ) is the finite part in d = 4 of the scalar integral in the M S scheme. The last term of (7.3), generated by the addition of a non-conformal sector (∼ n G ), vanishes separately as → 0. Finally, combining all the terms we obtain the regulated (reg) expression of the T T around d = 4 in the form The divergence in the previous expression can be removed through the one loop counterterm Lagrangian (2.67). In fact, the second functional derivative of S count with respect to the background metric gives having used the relation V µ 1 ν 1 µ 2 ν 2 E (p, −p) = 0. In particular, expanding around d = 4 and using again (7.2) we obtain which cancels the divergence arising in the two point function, if one chooses the parameters as b = − 3π 2 720 n S − 9π 2 360 n F − 18π 2 360 n G b = π 2 720 n S + 11π 2 720 n F + 31π 2 360 n G . (7.7) The renormalized 2-point function using (7.7) then takes the form In d = 4 this correlator manifests divergences in the forms of single poles in 1/ε (ε = (4 − d)), as for any CFT affected by the trace anomaly. These divergences are present in both the transverse-traceless and longitudinal parts. As discussed in detail in [37], the counterterm (2.67), for the 3-point case, renormalizes the correlator (7.9) by canceling all the divergences with the same choice of the coefficients (7.7) but, at the same time, gives extra contributions in the final renormalized T T T from the local/longitudinal part. These extra contributions defines the anomalous part of the correlator. In the case of n = 4 (2.62) specializes in the obvious form The anomaly part is given as which is the expression depicted in Fig. 1. The equation above has indeed a clear and simple interpretation in terms of anomaly poles extracted from the π µν projectors attached to each of the external graviton legs. We can re-express it in the form anomaly + (perm.) anomaly + (perm.) anomaly , (7.14) from which it is clear that a contribution such as where we trace one of the three stress energy tensors, is obtained by differentiating the anomaly functional twice. Similar results holds for the other contributions with double and triple traces It is clear from this result that all the possible anomaly contributions generated in the flat limit are associated with single, double and triple traces of the 3T, where on each external graviton leg we are allowing for a scalar exchange due to the presence of the transverse π projector, as clear from Fig. 1. This result is strongly reminiscent of the emergence of an anomaly pole in the AVV chiral anomaly diagram for a J 5 JJ correlator, with one axial vector (J 5 ) and two vector (J) which basically follows a similar pattern. A quite direct analogy with the behaviour of the T T T is present if we decompose the AAA anomaly diagram as 1/3(AV V + V AV + V V A), using its obvious symmetry. Notice that this result, for the TTT, is a consequence of the renormalized CWI's and can be deduced without any reference to perturbation theory. In both cases one encounters a scalar mode, via a bilinear mixing term attached to the external graviton lines, which is directly coupled to the anomaly, and is identified by the procedure of renormalization. Notice that the spin-2 part of the gravitational fluctuations do not couple to such bilinear mixing term, which mediates only spin-1 and spin-0 interactions. In the chiral case a similar (pseudo)scalar mode is present, interpreted as an effective (composite) axion. Therefore, we encounter a feature that unifies both the conformal and the chiral cases. It is also clear that this massless scalar interaction, in the TTT, is not removed by the inclusion of other Weyl invariant terms present, which are not identified by our method. For this correlator the anomalous trace WI takes the form

The four-point function
We will now come to illustrate the reconstruction procedure for the renormalized T T T T , showing how the separation of the vertex into a transverse-traceless part, a longitudinal one and an anomaly contribution takes place after renormalization. Clearly, by construction, the transverse traceless sector of the 4T is renormalized by adding the contribution coming from the counterterm If we consider a Lagrangian realization, the renormalization of this part and the corresponding form factors are ensured by the choice of the coefficients b and b as in (7.7), where n I , I = S, F, G, are the number of scalar, fermion and gauge fields running into the virtual corrections T T T T . For a general CFT, not directly related to a specific free field theory realization, the b and b should be interpreted as fundamental constants of that theory, and are arbitrary. Now, we turn to the longitudinal part of the correlator, which is the most interesting component when the case d = 4 is considered, due to the appearance of the anomaly. For instance, we study the bare part t loc T T T in (5.35) that is explicitly written as This contribution in d = 4 manifests some divergences due to the presence of the 3-and 2-point functions on its rhs. A similar equation holds for the counterterm in (3.11), which can be decomposed as well into the transverse-traceless part and the longitudinal one. The contribution that renormalises (7.20) is where we have taken into account the definition (2.67). It is worth mentioning that near d = 4 one has to use the expansion of the d-dimensional counterterms for the Weyl squared part as 22) and the expansion of the extra term present in (7.21) gives After adding the counterterms to the bare correlator and expanding around d = 4 we obtain anomaly (7.24) with the inclusion of an extra anomalous contribution in the final expression. In particular, this contribution takes the explicit form It is worth noticing that the anomaly of the 3-point function (7.13) contributes to the anomaly part of the 4-point function in (7.25), as expected. The same equation can be written in the simpler form and having taken into the anomaly part of the 3-point function defined in (7.14).

(7.34)
Finally, the last term that involves four insertions of the operator t loc , after renormalization takes the form t µ 1 ν 1 loc t µ 2 ν 2 loc t µ 3 ν 3 loc t µ 4 ν 4 loc bare + t µ 1 ν 1 loc t µ 2 ν 2 loc t µ 3 ν 3 loc t µ 4 ν 4 loc count d→4 = = t µ 1 ν 1 loc t µ 2 ν 2 loc t µ 3 ν 3 loc t µ 4 ν 4 loc (d=4) anomaly , (7.35) generating an extra term that contributes to the anomaly part of the full correlator as It can be re-expressed as In summary we obtained for the renormalized 4T the general expression anomaly (7.38) where the anomaly part can be identified using all the results presented above as anomaly . (7.39) We have shown how the anomaly part of the T T T T is extracted through the procedure of renormalization. It is clear from this procedure that such component is exactly the one predicted by the 4-dimensional reconstruction method, using the anomalous Ward identities.
In summary, we write the anomaly part of the correlator in the form (7.40) where the first contribution is anomalous (Weyl-variant) and the second contribution is traceless We call it the "zero residue" or the "zero trace" (0T) part, since the operation of tracing the anomalous part, removes the anomaly pole in the bilinear mixing terms, leaving a residue which is proportional to the anomaly. By analogy this part carries no pole. This contribution is explicitly given by the expression On the other hand, the anomaly part is then explicitly given as The picture that emerges from our analysis is depicted in Fig. 3 and is clearly the natural generalization of what found in the case of the 3T. We have shown though, that differently from the 3T, in the casse of the 4T there is an extra, Weylinvariant component in Γ A also predicted by the reconstruction. It is also clear why this component is not present in the 3T. Indeed, as discussed in Section 7, there is no anomaly in the T T and this is sufficient to set this extra contribution, predicted just from the CWIs, to vanish for 3-point functions.
Our result for the anomaly action, as predicted by the CWIs, can then be collected into the form where we have also included the (complete) T T , plus the extra traceless (0T) term appearing in the 4T, as identified in (7.42).
It is quite clear from (7.44) that the result can be organized at each order in the expansion in the gravitational fluctuations h in terms of pole parts and of traceless contributions. In principle, all the traceless contributions can be omitted from the definition of Γ A and the entire result can be expressed uniquely in terms of contributions affected by bilinear mixings (pole) terms. As we have emphasized at various stages in this work, we don't allow massless tadpoles in our regularization scheme, and the linear terms in h, which otherwise would be present and dominant at the Planck scale, in a flat background are absent. These terms would be phenomenologically important if one were interested in extending our analysis to the cosmological case. For instance, one possible application would be to determine the contribution of the conformal anomaly both in the early and late stages of the cosmological evolution, addressing the issue of the dark energy dominance of its more recent epochs. This can be performed using an extension of this procedure to more general backgrounds, strating from the Weyl-flat case. Also, the structure of the traceless contributions beyond the fourth order, induced by the CWIs at the level of the 5T and higher, can be worked out order by order, following the approach that we have illustrated, as we are going to describe in a related work.
Being the analysis formulated in the Euclidean case, it is clear that the extension of our result to Minkowski space requires an analytic continuation. It is quite clear, though, that this turns out to be trivial to perform, since the residui at each pole are purely tensor polynomial in momentum space and do not involve any analytic continuation around branch cuts. The residui computed at each pole, for any nT, are just given by anomaly polynomials, obtained by functional differentiations of the anomaly functional to arbitrary high orders. It is then clear that the anomalous CWIs predict the emergence of intermediate massless states coupled to such anomaly polynomials and build up the entire anomaly action in a quite simple and direct way.

Comments
The analysis of conformal anomaly effective action (Γ[g]) is important from many points of views, from the study of ordinary CFTs, explicitly broken by the trace anomaly, to cosmology and, in particular, in the quest for the nature of dark energy. For this reason, the characterization of correlators of stress energy tensors is of invaluable significance and deserves attention from both a theoretical and a phenomenological perspective. As we have pointed out in this work, the most direct way to investigate the structure of Γ[g] is to start from its definition as the generating functional of all the 1PI graphs obtained by integrating out conformal matter in the functional integral. It is therefore expressed as the sum of 1-loop amplitudes with an arbitrary number of external graviton lines. The direct computations of this action are hampered by the increasing complexity of the correlators of higher ranks involved, but the resummation of such contributions can be achieved, at least in part, by the integration of the anomaly equation, as for the Riegert action [72] or the Wess-Zumino actions with an extra dilaton, as in [55]. However, testing the prediction of the integration procedure against alternative, and, in our case, direct evaluations of the correlation functions involved in its expansion, is crucial in order to validate such class of actions. Different effective actions may by traceless contributions and, therefore, are not uniquely determined by the integration procedure. For this reason, the direct analysis of the corresponding correlators is one step forward in their characterization. They are investigated by embedding a certain CFT in a generic curved background, and computing its quantum averages using standard procedures.
One may consider various cases, the flat, the conformally flat and probably even more general backgrounds, in order to identify some of this constraints on the structure of such correlators.
In this work we have focused our analysis of the fluctuations, more simply, in the flat spacetime limit. In this case there are some obvious simplifications that take place, such as the vanishing of the entire stress energy tensor at first order in the flutuations h, using DR, and with the anomaly contribution appearing only from the 3-point function on. The analysis of CFTs in momentum space and the anomaly action is crucially related to the counterterm Lagrangian used for its regularization, and in the flat spacetime limit its renormalization, in d = 4, is performed only by introducing dimension 4 counterterms. Indeed, a CFT, as for any ordinary field theory in flat space, does not required any gravitational background to be correctly defined, and its renormalization can, in principle, proceed directly from flat space. For this reason, the curved background chosen in the evalutation of Γ should be viewed just as a convenient approach. In previous complete analysis of the TTT vertex, by going over the entire renormalization process of the form factors which appear in the transverse traceless part of this correlator, one reaches some important conclusions concerning the structure of the anomaly contribution of such vertex. In the TTT, this is characterised by the presence of bilinear mixings which signal a broken phase induced by the anomaly. Such mixings can be clearly interpreted as due to the generation of an interpolating state, as shown in (7.14), depicted in Fig. 1, in the form of a sequence of bilinear graviton-spin 0 mixing terms which had been already identified in perturbation theory in several previous analysis. These terms describes effective dilaton interactions induced in a vacuum state of a CFT induced by the renormalization process.

Conclusions and Perspectives
In this work we have shown that a previous analysis of the the TTT vertex, which took to a special characterization of the anomaly action through cubic order in the gravitational fluctuations, can be extended to the 4-graviton vertex TTTT. This has been obtained exclusively by a careful analysis of the CWIs that such correlators have to satisfy. For this purpose, we have extended to the 4T the longitudinal/transverse decomposition, introduced in the case of the 3T, which is ideal for a detailed analysis of the conformal constraints. The method is pretty general, and can be extended, as we are going to show in a separate work, to general dimensions. We have also shown that these results can be obtained without the need to proceed with a complete analysis of all the form factors which appear in the transverse traceless sector of such correlator. We have shown that the anomaly action carries a very specific signature of the breaking of the conformal symmetry by the conformal anomaly, in agreement with a former analysis of the same action at cubic level. We have also shown that, respect to the case of the 3T, the conformal constraints indicate also the presence of a new traceless and separately conserved component. There are several possible implications of these results, such as in gravitational waves [73] and in solid state [74][75][76], which have been performed using the Reigert action, which predicts a structure of the 2T and 3T in agreement with the analysis presented here, and can be investigated, in parallel, using our formulation. Our approach can be generalized in several directions, for instance by considering fluctuations around a conformally flat space, where the dynamics of the dilaton on the renormalization process should emerge more clearly compared to the case discussed here. Indeed, the method is very general, since it allows, in principle, to characterize the anomaly action to any arbitrary order, providing an alternative to the Weyl gauging procedure used for the determinaton of such action. We hope to address some of these new important issues in future work.

A Contact terms
Contact terms generated by 4th variation of a fundamental action are generated in the form δT(a) δg(d) In the flat limit we obtain .