Four-point functions of gravitons and conserved currents of CFT in momentum space: testing the nonlocal action with the TTJJ

We present an analysis of the perturbative realization of the TTJJ correlator, with two stress energy tensors and two conserved currents, using free field theory, integrating out conformal sectors in the quantum corrections. This allows to define, around flat space, an exact perturbative expansion of the complete anomaly effective action – up to 4-point functions – whose predictions can be compared against those of the anomaly induced action. The latter is a variational solution of the conformal anomaly constraint at \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d=4$$\end{document}d=4 in the form of a nonlocal Wess–Zumino action. The renormalization procedure and the degeneracies of the tensor structures of this correlator are discussed, valid for a generic conformal field theory, deriving its anomalous trace Ward identities (WIs). In this application, we also illustrate a general procedure that identifies the minimal number of tensorial structures and corresponding form factors for the TTJJ and any 4-point function. The approach is implemented for three, four and five dimensions, addressing the tensor degeneracies of the expansion in momentum space. We show that the renormalized TTJJ can be split into two contributions, a non anomalous and an anomalous part, each separately conserved. The first satisfies ordinary trace WIs, while the second satisfies anomalous trace WIs. The result of the direct computation is compared against the expression of the same 4-point function derived from the nonlocal anomaly induced action. We show that the prediction for the anomalous part of the TTJJ derived from such action, evaluated in two different conformal decompositions, the Riegert and Fradkin–Vilkovisky (FV) choices, differ from the anomaly part identified in the perturbative TTJJ, in the flat spacetime limit. The anomaly part of the correlator computed with the Riegert choice is affected by double poles, while the one computed with the FV choice does not satisfy the conservation WIs. We present the correct form of the expansion of the anomaly induced action at the second order in the metric perturbations around flat space that reproduces the perturbative result.


Introduction
The study of multi-point tensor correlators in conformal field theory (CFT) plays a significant role in several contexts, with applications to nonlocal cosmology, condensed matter theory and particle phenomenology. Conformal symmetry describes the properties of systems at energy scales at which the dynamics of the correlation functions are constrained by the generators of the conformal group, whose number varies according to the spacetime dimensions (d). In d = 4, a conformally invariant theory is characterized by a partition function that satisfies all the 15 constraints coming from the conformal group SO (2,4), modulo the presence of conformal anomalies. Conformal anomalies [1] are associated, for a given CFT, with the need to renormalize the theory, according to a regularization procedure which, at least around a flat spacetime, can be implemented directly using ordinary Dimensional Regularization (DR). Indeed, all the conformal constraints can be derived, for Lagrangian CFTs, by resorting to the formalism of the effective action via its functional expansion with respect to the external background metric [2,3]. In the pure gravity sector, its renormalization can be performed using only two counterterms: the Euler-Poincarè density (E) (Gauss-Bonnet) and the Weyl tensor squared (C 2 ). Both terms will play an important role in our analysis. All the quantum corrections and the conformal constraints can be generated by investigating such renormalized partition functions in the form of trace and special conformal Ward identities either in coordinate [4,5], or in momentum space [6,7] [8,9]. The breaking of scale invariance by the renormalized correlator is present in its traceless part. The derivation of such constraints is general and formulated in curved spacetime, and subsequently can be specialized around flat Minkowski space. This approach bypasses the traditional operator formalism used in deriving the same constraints in the standard CFT literature. The analysis around flat space allows to address quite clearly several issues left open by the renormalization procedure, such as the breaking of scale invariance due to the inclusion of a renormalization scale, which is missing in both local and nonlocal effective actions [10] of Wess-Zumino forms, commonly discussed in the anomaly literature. Scale violations are expected to be part of the complete effective action, which is at the centre of our work. However, they are not present in those actions identified as possible variational solutions of the anomaly constraints. These do not necessarily include the quantum corrections discussed in this work, that will be taken into account for the T T JJ correlator. While all the correlators of a certain CFT are of interest, some play a crucial role in the possible implications of such theories in realistic physical contexts. Conformal anomaly actions, investigated in their Weyl-invariant and anomaly terms, provide definitive information about the correct effective field theory description that results once a conformal sector is integrated out of a specific partition function. This procedure modifies the background gravitational metric in a very special way. The extra terms induce a gravitational backreaction that can be studied around flat space with great accuracy and quite explicitly in specific renormalization schemes [10].

Implications for topological matter and nonlocal cosmology
Among all the correlators, those containing stress-energy tensors and classically conserved vector and chiral currents, besides the scalar CFT primaries, for the reasons mentioned above, are certainly among the most important ones. They can be studied in ordinary field theories of conformally coupled scalars, fermions and spin-1 gauge fields in free field theory realizations [11][12][13] [14,15]. Such realizations match exactly, at least for those correlators containing stress energy tensors (T ) and conserved currents (J), the general tensor structure predicted for them by the conformal Ward Identities (CWIs) of a CFT, with significant simplifications [16][17][18]. In condensed matter theory, they find application in the study of topological materials, such as Dirac and Weyl semimetals, where anomalies are expected to play an important role. Due to the presence of an effective linear dispersion relation in the band structure of such materials and to their topological character (see [19][20][21][22][23][24]), chiral [25,26] and conformal anomaly actions can be used to describe their response functions and anomalous transport under external sources. One link, for example, is provided by Luttinger's formula, that relates a thermal solicitation of such materials to an external gravitational field [23,27]. A second important sector in which these analyses play a role is in the physics of the early universe and the production of gravitational waves for defining special corrections to gravity which may either take the form of local scalar tensor theories or a nonlocal form (see for instance [28,29]).

Momentum space analysis
The constraints imposed by the conformal group for d > 2, up to 3-point functions, are sufficient to identify the correlation functions of a certain CFT only modulo few constants [4]. The constants appearing in the general solutions of correlation functions, such as the T T T or T JJ, are reproduced by combining a certain number of sectors with arbitrary particle multiplicities (n S , n f , n V ) of scalars, fermions and gauge fields, respectively. We recall that for higher point functions, CWIs do not predict the exact form of such correlators, since arbitrary functions of the conformal invariant ratios -which depend on their coordinate points -are part of their general expressions. This arbitrariness has also been discussed in momentum space [30], at least for scalar correlators. The analysis of tensor correlators is far more involved and has been formulated in momentum space for 3-point functions of T 's J's and scalar operators in [7]. However, it remains valid also for n-point functions [3,31] (see [8,32]), covering also the Minkowski signature [33][34][35] as well as applications to cosmology [36][37][38][39]. We will also be using the same approach in our case, similarly to the case discussed in [3], but working directly with a free field theory realization. A lot of insight and essential information about the structure of correlation functions can be uncovered by working directly in momentum space [16]. Indeed, one of the limitations of CFT in coordinate space is the difficulty of describing the implications of the conformal anomaly in a complete way and, in particular, the anomaly action. In this approach, the anomaly is introduced by hand, in each correlation function, by extending the solutions of the (non anomalous) CWI's with the addition of ultralocal terms. For the rest, the anomaly contributions are absent in any application based on the operator product expansion (OPE). The OPE is an operatorial expansion at short distances that necessarily avoids spacetime regions where all the points of a certain correlator coalesce. On the other end, the momentum space analysis allows us to derive the anomaly contributions of a certain correlator in a very natural way since integration over momentum space obviously also covers the contact regions in the external coordinate points of the correlator.

The partition function
We start our discussion by defining the unrenormalized partition function of the theory Z B (g), identified by the bare functional Z B (g) = N Dχ e −S 0 (g,χ) , (2.1) where N is a normalization constant. We have denoted by χ, a collection of conformal fields that in d = 4 correspond to scalars, fermions. Our analysis is set in the Euclidean case. The transition to Minkowski space can be performed by a simple analytical continuation of the correlation functions, since we will be dealing with free field theory realizations. The bare effective action will be defined as Quantum matter fields are assumed to be in a conformal phase at d = 4. The bare effective action S B (g) includes all the multiple insertions of the stress energy tensor (pure gravity sector) and mixed graphs with photons and gravitons.
As usual, Z B (g), which is the semiclassical effective action (see for instance [40]) in the Feynman diagrammatic expansion, will contain both connected and disconnected graphs, while S B (g) collects only connected graphs.
In the gravitational sector, the expansion provides all the pure graviton vertices, the mixed graviton/gauge and pure gauge vertices, defined by the insertions of the stress energy tensor and of the gauge current J µ . For instance, the quantum averages of 1-point functions are defined as where the metric is taken to be flat after the variation. Similarly, correlation functions of higher order are defined in a metric backgroundḡ and with a vanishing gauge field A µ , by varying both external fields S(g) B ≡ S(ḡ) B + ∞ n=1 1 2 n n! d d x 1 . . . d d x n √ −g 1 . . . √ −g n T µ 1 ν 1 . . . T µnνn ḡB δg µ 1 ν 1 (x 1 ) . . . δg µnνn (x n ), + ∞ n,k=1 1 2 n n!k! d d x 1 . . . d d x n d d x n+1 . . . d d x n+k √ −g 1 . . . −g n+k T µ 1 ν 1 . . . T µnνn J µ n+1 . . . J µ n+k ḡB × × δg µ 1 ν 1 (x 1 ) . . . δg µnνn (x n )δA µ n+1 (x n+1 ) . . . δA µ n+k (x n+k ) + . . . (2.4) where the dots refer to contributions from the pure gauge sector in the expansion. The covariant normalization of the correlation functions is given by for the n-graviton sector, with with √ −g 1 ≡ −|det g µ 1 ν 1 (x 1 )| and so on, and by for the graviton/gauge sector. Diagrammatically, the pure gravitational sector is identified, in free field theory realizations, by an infinite sum of 1-loop diagrams with an arbitrary number of external graviton lines. The mixed sector will include the T T J, the T JJ and the T T JJ correlation functions. The diagrammatic expansion of the T T JJ for the fermionic and scalar cases are given below in Fig. 6 and Fig. 7. In Dimensional Regularization (DR) the renormalized effective action is defined by the inclusion of three counterterms where N is a normalization constant, = d − 4, and are the counterterms corresponding to the Gauss-Bonnet and Weyl tensor squared densities and the Weyl terms respectively. In order to remove these divergences of the mixed graviton/gauge correlators we add to the action the counterterm corresponding to the field strength F 2 = F µν F µν where the coefficients β c (I) refer to the scalar and fermion contributions.

Local and nonlocal actions and the T T JJ test
Most of the analysis of anomaly actions in the literature deals with the problem of the identification of a functional whose variation with respect to the metric generates the conformal anomaly. The correlation functions extracted from the renormalized action satisfy hierarchical CWIs that allow the identification of two contributions, denoted as the Weyl variant and the Weyl invariant parts, respectively. As just mentioned, in the case of pure graviton vertices, the Weyl variant part of the renormalized effective action is related to the two counterterms V E and V C 2 , while in the T T JJ a third counterterm, V F 2 , is needed. Their variations reproduce the anomaly since while the bare effective action, corresponding to the Weyl invariant part, in d dimensions satisfies the condition The Weyl variant part can be summarized by the functional i.e. the counterterm action (with β C = I=f,s n I β c (I)) and this separation between S B and S v , with is perfectly well defined as far as d = 4. S B becomes singular at d = 4 and the renormalization procedure consists in expanding around four spacetime dimensions both S B and the counterterm. The expansion is performed using a fiducial metricḡ and its fluctuations, withḡ taken to be the Minkowski metric. Using the fact that the singularities of S B are removed by the singular parts of the counterterms, S R gets effectively re-expressed in the form and the finite contribution coming from the loops contained in S f The anomaly action generated by this regularization can then be defined in the form In standard approaches in which one tries to solve the constraint ignoring the renormalization process implicit in the extraction of the effective action at d = 4, which requires to take a singular limit, the classification of S R in terms of the two parts does not introduce any extra scale. Obviously, this simply implies that we are focusing only on S v in (2.15), neglecting, at the same time, the presence of extra scales generated by the virtual quantum corrections, due to the renormalization procedure. The idea of using the realizations of free field theory in flat space, as already mentioned, allows us to have a firm grip on the structure of the expansion, although this is only possible for a simpler background, compared to a curved one For these reasons, it is not surprising that a functional solution of the anomaly constraint, which corresponds to an anomaly induced action, may not reproduce the perturbative result and the Ward identities that come with it.
As we are going to show by an explicit computation, there is a perfect agreement between the complete effective action derived by a free field theory realization and the anomaly induced actions that we are going to discuss next, up to 3-point functions. The two actions are derived by selecting two different conformal decompositions, usually termed "gauge choices" in the literature, in which the dilaton field is expressed in terms of the metric g by two different functional constraints. In our example, they correspond to the Riegert (Σ R ) and the Fradkin-Vilkovisky (Σ F V ) choices. We are going to investigate this point performing a direct computation on the two actions, showing that as we move to 4-point functions, the correlators do not satisfy the hierarchical Ward identities of the case.

Ward Identities
The symmetry constraints on S R , induced on the coefficients of the expansion (2.4) take the form of WIs which are hierarchical. The conformal constraints, for instants, are linked to the Weyl invariance of the renormalized effective action and to its breaking and a derivation of the corresponding WIs can be performed directly from either S B or S R , as shown in [2,3]. We recall that in a curved background, for a certain action S(g), Weyl invariance is expressed as a symmetry of the form The relation between g andḡ defines a conformal decomposition, which remains valid under the gauge transformationḡ where σ(x) is a local shift. The renormalization of the quantum corrections, via the counterterms above, breaks this symmetry. In the case of a flat background, one is essentially performing the φ → 0 limit of S R after performing the metric variations, with the dilaton variation δ δφ replaced by 2g µν δ δgµν . A study of the semiclassical effective action in the presence of a dilaton background is in [41]. In general, on the bare functional S B (g), one derives the relation and its invariance under Weyl δ φ g µν = 2g µν δφ, (3.4) and diffeomorphisms δ g µν = −∇ µ ν − ∇ ν µ , (3.5) are summarised by the constraints leading to trace and conservation conditions of the quantum averages of T µν Ordinary trace and conservation WI's can be derived from the equations above by functional differentiations of S B (g) with respect to the background metric. As far as we stay away from d = 4 and include in the classical action S 0 conformal fields, we have exact CWI which are derived from the condition of invariance of the generating functional S B with respect to diffeomorphisms and Weyl transformations. Anomalous CWIs are derived by replacing the effective action S B with the renormalized one S R . Non conformal sectors, such as spin-1 contributions, modify the CWIs by inhomogeneous terms unrelated to the anomaly, which is a pure 4d phenomenon. We move to discuss the derivation of the conformal and conservation WIs for the correlator. This allows us to illustrate its decomposition, following the approach of [7], in terms of a transverse traceless sector, a longitudinal sector and a trace sector. Only the trace and conservation WIs will play a role in our analysis.
Assuming that the generating functional of the theory is invariant under the action of some symmetry groups, then the correlation function T T JJ satisfies with G g the generators of the infinitesimal symmetry transformations. These constraints come from the invariance of the generating functional under symmetry transformations that can be expressed equivalently as Among these constraints, the conservation Ward Identity (WI) in flat space of the energy momentum tensor can be obtained by requiring the invariance of S B [g, A] under diffeomorphisms x µ → x µ + µ (x) for which the variation of the metric and the gauge fields are the corresponding Lie derivatives. In the case of a nonabelian SU (N ) gauge field A a µ (a = 1, 2, . . . , N 2 − 1), for instance, we get Inserting these variations into (3.10) and integrating by parts, we obtain the conservation WI Similarly, the requirement of invariance under a gauge transformation with a parameter θ a (x) gives δg µν = 0, (3.15) and the invariance of the generating functional under gauge transformations gives Inserting this equation into (3.13) we obtain the conservation WI's In the Abelian case, which is the case of our interest, diffeomorphism and gauge invariance give the relations By functional differentiations of (3.18) we derive ordinary WIs for the various correlators involving energymomentum tensors and conserved currents. In the T T JJ case we obtain, after a Fourier transform, the conservation equation where the notation (3 ↔ 4) means the exchange of the subscript 3 with 4, and the vector current Ward identities In our conventions, all the momenta, in a given correlator, are incoming. Furthermore we consider the invariance of the generating functional under Weyl transformations for which the fields transform as in (3.4) and giving the naive trace Ward identity The functional differentiation of (3.22) gives the (non-anomalous) condition that is preserved for d = 2n, n ∈ N. In Appendix A we offer more details on the conservation WI's.

Decomposition of the correlator
As already mentioned, the general form of the T T JJ correlator can be constructed by a decomposition into transverse, longitudinal and trace terms [7], exploiting its symmetries. We start by decomposing the operators T and J in terms of their transverse traceless part and the longitudinal (local) ones having introduced the transverse-traceless (Π), transverse (π), longitudinal (Σ) projectors, given respectively by By using the projectors introduced above, the correlator can be written as and It is clear now that only the second term in (4.8), expressed explicitly in (4.11), will contain the entire trace and longitudinal contributions, with these two sectors constrained by the conservation WIs (3.19), (3.20) and (3.23). Thus, the unknown part of the correlator is contained in its transverse-traceless (ttjj), since the remaining longitudinal + trace contributions -the local terms -are related to lower point functions by conservation and trace WIs. Therefore, we can proceed by studying the general decomposition of the transverse-traceless part ttjj into a product of form factors and tensor structures. Due to conditions (4.9) and (4.10), such sector, using the transverse and traceless projectors, can be written in the form where X α 1 ...α 4 is a general rank six tensor built out of products of metric tensors and momenta with the appropriate selection of indices. Note that due to the presence of the projectors in (4.12), the terms δ α i β i , i = 1, 2 or p α i i , i = 1, . . . , 4 cannot be used as fundamental tensors and vectors to build the X α 1 ...α 4 tensor. In addition, the conservation of the total momentum allows to select for each index α i a pair of momenta out of the four, to be used in the general construction of X.
The choice of the independent momenta of the expansion, similarly to the case of 3-point functions discussed in [7], can be different for each set of contracted tensor indices. We will choose as a basis of the expansion, for each pair of indices shown above. Once the decomposition has been performed according to this scheme and the number of form factors identified, one momentum, for instance p 4 , can be chosen as the dependent one. This approach is rather economical, since it allows to reduce the number of form factors to a minimum, exploiting the presence of a single tt projector for each external momentum. Regarding the tensor structures built out of metric δs, the only non vanishing ones appearing in X α 1 ...α 4 are together with the similar ones obtained by the exchange α i ↔ β i , i = 1, 2. This strategy has been introduced in [7] for 3-point functions and applied also to the case of 4-point functions in [3]. In the next section, we are going to describe the procedure in order to write explicitly the expression of X α 1 ...α 4 in terms of the minimal number of tensor structures and form factors, in general d dimensions [31].

Orbits of the permutations
X α 1 ...α 4 is expressed in terms of tensor structures and form factors using the symmetry of the correlator. The T T JJ manifests two types of discrete symmetries related to the permutation group: it must be symmetric under the exchange of the two gravitons (1 ↔ 2), of the two conserved J currents (3 ↔ 4), and the combination of both transformations. We label such transformations respectively as P 12 , P 34 and P C = P 12 P 34 . It is worth mentioning that P 12 exchanges the pair of indices of the two gravitons and the momenta associated with them, and analogously for the two currents J's. The tensorial structures in X α 1 ...α 4 will be constructed by using the metric tensors and the momenta with the choices (4.15) and (4.14). Then, in X α 1 ...α 4 , there are structures of four different type, depending on the number of metric tensors and momenta used to saturate the number of free indices. We consider the general terms δδδ, δδpp, δpppp, pppppp, (4.16) observing that these sectors do not mix when the permutation operator P ij is applied. This property allows us to construct the general symmetric form of each sector separately. As a first step, we determine the orbits of the P operators acting on the tensor structures belonging to each tensorial sector (4.16). This can be achieved by applying all the P transformations to a tensor structure and following the "trajectory" (orbits) in the sector generated by this process. For instance, in the sector δδpp, we encounter the two orbits In this way we decompose every sector (4.16) into orbits. Every P transformation acts on an orbit irreducibly, i.e. it connects every element on the orbit. The number of orbits for all the sectors equals the number of independent form factors representing the correlator. In fact, a representative can be selected for each orbit to which an independent form factor can then be associated. The orbit provides a visual realization of the symmetry properties of the form factors that belong to it. For clarity, let's clarify this statement with an explicit example, using the sector δδδ. With the choices (4.15) and (4.14) we construct the three possible tensor structures in X α 1 ...α 4 as This sector can be decomposed into two orbits as from which we can conclude that for this sector there exist just two independent form factors related to the representatives of the orbits. Then the general form of X for the sector with three δs (denoted as zero momentum or "0p") can be directly written as where the two independent form factors satisfy the symmetry conditions The properties of the orbits within the tensorial sector are directly reflected in the symmetry conditions of the form factors.
The method proposed in [31] is similar, but more involuted, since it starts from the most general tensor structure with a non-minimal number of form factors. By imposing the symmetry conditions under the group of permutations, one obtains the conditions that reduce the number of form factors and the symmetry constraints that they have to satisfy. Indeed, applying the prescription of the example above, we should identify its expression starting from the general ansätz The invariance of the correlator under the permutation P 12 reduces the number of independent form factors and gives the symmetry conditions The invariance of the correlator under the other symmetry transformations P 34 and P C , turns into some symmetry conditions on the independent form factors as obtaining again (4.21). It is now clear that the study of the orbits of the tensor structures under the permutation group, provides directly the answer about the minimal number of independent form factors that are needed in order to describe the general solution of any 4-point correlator. This procedure can be simply generalized to higher point correlation functions involving operators of any spin. Once we identify and select representative of each orbit for every sector, then the general structure of X α 1 ...α 4 can be written down quite easily. In this way, we find that in d > 4 the general form of X α 1 ...α 4 related to T T JJ is written in terms of 47 independent form factors. This number reduces significantly when d ≤ 4, as we are going to show in the following sections. The number of tensor structures and independent form factors for the general d dimensional case is listed in Tab. 1 and the representative of each orbit are listed in Appendix B.  5 Dimensional dependent degeneracies of tensor structures in d ≤ 5 In the previous section we have presented the method to construct the general form of the correlator in order to obtain a minimal number of form factors in the general d dimensional case. In d ≤ 5, the structure of X α 1 ...α 4 changes, according to the degeneracies of the tensor structures [3,9,42,43]. These degeneracies cause a reduction in the number of independent form factors and significantly simplifies the structure of the correlator. In this section, we discuss the dimensional reduction patterns for d ≤ 5.

Case of d = 5
Following the argument presented in [42,43], for any tensor in d-dimensional space there is associated a fundamental tensor identity obtained by antisymmetrizing over d + 1 indices. In particular let be a trace-free tensor on all of its indices, where A denotes an arbitrary number of additional lower and/or upper indices. Then where n ≥ d − k − l + 1 and n ≥ 0.
In d = 5, (5.1) admits a tower of tensor identities depending on the values of k and l with the condition n ≥ 6 − k − l and n ≥ 0 as showed in Tab. 2. It is worth mentioning that for l + k ≥ 6 we obtain the condition that the tensor itself vanishes for n = 0, and all the other identities for n > 0 are trivially satisfied. In order to clarify this point, let us consider the case with k = 3, l = 3 for which, in d = 5, the relation (5.1) becomes The first identity, depending on the values of n, is obtained for n = 0 k l n Tensor rank k l n Tensor rank Table 2: Possible tensor identities depending on the relation n ≥ 6 − k − l.
which trivially satisfies all other identities for n > 0. For example, using the fact that an anti-symmetrization on n indices can be decomposed iteratively on a smaller number of indices, in the case n = 1 one obtains by means of (5.3). After a meticulous investigation of all the tensorial identities in d = 5, we present below only those that affect the form of the tensorial structure of the T T JJ. In particular, all the constraints from Tab. 2 are equivalent to only one. To show which one is the constraint for the T T JJ in d = 5, we consider the case with k = 3 and l = 3 that takes the form In order to construct the antisymmetric tensor T , we consider the tensor and then the traceless and completely antisymmetric tensor T is directly written in the form This is the only non-trivial tensor that can be constructed using the three independent momenta of a 4-point function. The use of the metric δ b i a i in (5.6) gives a trivially zero result in five dimensions. In order to find the dimension-dependent equation of the tensor structure of the transverse-traceless part of the T T JJ , we apply to (5.7) the Π and π projectors From this analysis, we conclude that in d = 5 the number of independent form factors is reduced by one according to the constraint (5.9) on the tensor structures.

Case of d = 4
Analogously to the previous case, in d = 4 one has to consider tensor identities derived from (5.1). However, in this case, we are going to show that there is an even more efficient way to identify the independent tensorial structures. Indeed, in d = 4, the metric δ is not an independent tensor and one can construct a different basis in which it can be re-expressed. For this purpose, we construct a new orthogonal four-vector n µ using the completely antisymmetric tensor µ 1 ...µ 4 and three of the four external momenta in the form Notice that this reduction is possible if one has in d dimensions at least d − 1 independent external momenta in a correlation function. The vector (5.11) is obviously transverse with respect to p 1 , p 2 , p 3 , i.e. n · p i = 0. Having defined such a vector, we use this new basis, that we call the n-p basis, to construct all the tensorial structures that we use to define the correlation function. In this new basis, the metric tensor is expressed as In particular, the Gram matrix in this case is written as from which we obtain the expression of δ as where the indices i, j are labelled mod-3 and With the expression for the metric given in (5.14), and using the constraint with the indices labelled mod-3. With the conventions for independent momenta and indices made in (4.14), the constraint can be explicitly rewritten as The previous constraints, occurring when at least two n vectors are contracted with a transverse traceless projector, allow us to write the decomposition of the transverse-traceless part (4.12) in d = 4 in terms of just three sectors, i.e. nnnnpp, nnpppp, pppppp.
It is worth to point out that in this new basis the term n α 1 n β 1 n α 2 n β 2 n α 3 n α 4 cannot appear in the decomposition of X α 1 ...α 4 , since because of the constraints (5.17) it would be rewritten as p α 1 i p β 1 j p α 2 k p β 2 l n α 3 n α 4 with i, j, k, l = 3, 4. Furthermore, in this new basis we notice a drastic reduction of the number of independent tensorial structures of (5.18). By analysing the orbits of the tensor structures under the permutation group, we select the representatives of each of them (see Appendix B) and thus determine the number of independent form factors in d = 4. The result is considerably simplified respect to the general case. We conclude that in this specific case the transverse traceless part of the T T JJ is parametrised by 34 independent form factors as summarized in Tab. 3.

Case of d = 3
In the case of d = 3 the decomposition of the ttjj reduces significantly. Indeed, also in this case the δ is not an independent tensor but it can be written in terms of the external momenta. Furthermore, contrary to the case of d = 4, there is no need to introduce an additional orthogonal vector. The three independent momenta p 1 , p 2  and p 3 are sufficient to span the new basis. The three dimensional version of (5.12) and (5.14) is immediate and we have where J is the determinant of the Gram matrix given as The form of the δ in three dimensions in (5.19) allows us to conclude that the only structure present in (4.16) is formed just by momenta, and we have to study a single tensorial sector made of six momenta. Also in this case we have some constraints coming from the contraction of (5.19) with a transverse-traceless projector. Indeed, the property By using these constraints we find that at d = 3 the number of tensor structures is smaller respect to the general case and that of d = 4. Following the procedure described in the previous sections, we find that the number of independent form factors in this case is reduced to five, as shown in Tab. 4.

Summary
We are ready to summarize the results obtained in the previous sections. By studying the orbits of the tensor structures under the group of permutations, we have identified the minimal tensor structures and form factors needed to construct the transverse-traceless part of the correlator. This study has been first performed in the case of d ≥ 6. For d ≤ 5, because of degeneracies in the tensor structures, one needs to take into account identity (5.1). In particular, for d = 5, we find one independent constraint that reduces the number of form factors by one. In d = 4, we have shown how to rewrite the metric δ µν in the basis n-p by introducing an orthogonal vector n given by (5.11). This approach can also be performed for d = 3, showing that the independent tensor structures are considerably reduced, as is the number of independent form factors. It is worth mentioning that when it is possible to define a new independent tensor base, the identities (5.1) are automatically satisfied. This property has been checked in the case of the T T J. The generalization of this approach to any 3or 4point correlator will be presented elsewhere. The number of minimal form factors for specific dimensions is summarised in Tab. 5.
T T JJ 47 46 34 5 where N p is the number of momenta multiplying the form factor in the decomposition. This equation characterizes the scaling behaviour of A (d),Np and allows to identify quite easily those among all which will be manifestly divergent in the UV regime. In general, their behaviour is summarized in Tab. 6. In all the cases, whenever divergences are present, they will show up as single poles in the regulator ε. The procedure of renormalization, obtained by the inclusion of the counterterm, will remove these divergences and will generate an anomaly. In the next section, we are going to discuss the appearance of the conformal anomaly after the renormalization procedure, for the d = 4 case.

Perturbative analysis in the conformal case
In this section we direct our attention towards an important aspect of our analysis, which will allow us to extend the result of the T JJ correlator presented in [18,44] to the T T JJ. The two correlators are connected via the hierarchical structures of the trace and conservation WIs and the renormalization procedure. We use the free field theory realization of the T T JJ in order to study the structure of the anomalous Ward identities once the conformal symmetry is broken after its renormalization. After the renormalization, the vertex will separate into a renormalized part of the form (4.8) and an anomaly part, following the same pattern of the T JJ. This anomaly part will be compared directly with the result coming from the variation of the anomaly effective action. We start our analysis by defining our conventions for the perturbative sectors. The fundamental classical action S 0 can be defined as a sum of two sectors, the scalar and the fermion sectors, which will be considered separately. It can be defined in the form where l = 1, 2 . . . N s and l = 1, 2 . . . N f enumerate the conformally coupled scalars and fermions in the total action. We will consider the case of a single scalar and a single fermion, correcting the results for their multiplicities at the end. The action for the fermion field in a gravitational background is where V µ a is the vielbein and V its determinant, D µ is the covariant derivative defined as where Σ ab are the generators of the Lorentz group in the case of a spin 1/2-field, and being the spin connection in the holonomic (metric) definition, with the antisymmetric property ω µ ab = −ω µ ba . The Latin and Greek indices are related to the (locally) flat basis and the curved background respectively. Using the explicit expression of the generators of the Lorentz group one can re-expresses the action as follows where γ abc is a completely antisymmetric product of gamma matrices defined as We recall the property Taking a first variation of the action with respect to the metric one can construct the energy momentum tensor as The computation of the vertices can be done by taking functional derivatives of the action with respect to the metric and the gauge field and Fourier transforming to momentum space. They are given in Fig. 1 and their explicit expressions have been collected in Appendix E. The action of a scalar coupled to a gauge field in a where R is the scalar curvature and φ denotes a complex scalar field and D µ φ = ∇ µ φ + ie A µ φ, the covariant derivative for the coupling to the gauge field A µ . The vertices for the scalar interactions depicted in Fig. 2 are listed in Appendix E. It is worth mentioning that we have not considered the vertex with two gauge fields/two scalars and two gravitons because it contributes as a massless tadpole at one-loop, and vanishes in DR.

Feynman Diagrams
In order to find all the Feynman diagrams that will contribute to the correlation function, we start from the definition of the energy momentum tensor as given in (2.3). The T T JJ correlator around flat space is extracted by taking four derivatives of the bare effective action S B with respect to the metric and the gauge field, evaluated when the sources are turned off (7.11) We will discuss the Euclidean case, with background metric δ µν . The analytic continuation of our results to Minkowski space are pretty straightforward, since the basic master integrals appearing in the computations p2 p1 p3 p4 l + p2 (c) Figure 3: Examples of box diagrams that contribute to the T T JJ correlator.
are the scalar self-energy, the triangle and the box diagram. Having denoted with S 0 the conformal invariant classical action in S B , we have where for sake of simplicity we have indicated for g i = g µ i ν i (x i ), i = 1, 2 and for A j = A µ j (x j ), j = 3, 4. In (7.12) the angle brackets denote the vacuum expectation value and all the terms are referred to a particular topology for the Feynman diagrams. In particular, we have the box diagram topology, the first term in (7.12), then the triangle diagrams topology expressed by the second term and the second line of (7.12), and finally the bubble diagrams written in the last line of the same equation. As we have already mentioned, we discard the term δ 4 S 0 δg 1 δg 2 δA 4 δA 3 (7.13) in (7.12), because it generates a tadpole diagram at one-loop. For the fermionic case, examples of diagrams are shown in Fig. 3. For instance, the first in Fig. 3 (a), is summarized by the expression Figure 4: Examples of triangle diagrams of the triangle topology.
A few additional examples of triangle diagrams contributing to the T T JJ can be found in Fig. 4. The first diagram, for instance, is expressed as Other topologies are given by the bubble contributions. Some examples of these are shown in Fig. 5. For instance, for the first diagram we have All the diagrams computed in the fermionic and scalar cases are given in Fig. 6 and Fig. 7. The perturbative realization of the T T JJ will be written down as the sum of these amplitudes, formally given by the expression where G is the set of all the diagrams listed in Fig. 6 and Fig. 7, and S i is the symmetry factor of each diagram. This perturbative realisation of the T T JJ in d dimensions satisfies the conservation and trace Ward identities (3.19), (3.20), (3.23). However, around d = 4, some loop integrals diverge and the renormalisation procedure of this correlator will lead to the appearance of an anomalous part (the trace anomaly). Scale-breaking p2 p1 p3 p4 l + p2 contributions will be naturally associated with the renormalization procedure, and they are not accounted for by the trace anomaly condition, as we will discuss in the next sections. From (7.17), by using the transverse π and transverse traceless Π projectors defined in (4.7) and (4.6), we obtain the perturbative realization of the transverse traceless part 18) and then identify the form factors described in Section 4. The computation of all the diagrams has been performed explicitly and the expressioon of the renormalized CWI's satisfied by the form factors of the T T JJ will be discussed separately for d = 4, since they are rather involved.

Divergences in d = 4 and renormalization
We have computed the explicit expressions of the form factors in Section 6 and classified the divergent ones, that coincide with the list given in Tab. 6. As already mentioned, their expressions are finite at d = 3. Here, in p2 p1 p3 p4 l + p2 this section, we focus on the analysis of the structure of their divergences in DR at d = 4. At d = 4, from Tab. 6, we expect that the form factors multiplying the tensorial structures with (2δ, 2p) and (3δ) are divergent. We actually find, form the perturbative calculations, that the divergent part of the transverse traceless component is written, after some manipulation, as where N f and N s indicate the the number of fermion and scalar families respectively, that are arbitrary. The projectors Π are expanded around d = 4 as with Π (4) the transverse traceless projectors defined in (4.6) with d = 4. The counterterm vertex is From this counterterm we can extract its transverse traceless part that can be written as where s = (p 1 + p 2 ) 2 . From (7.19) and (7.23) we notice that the divergences are removed with the choices β c (s) = e 2 96 π 2 , β c (f ) = e 2 24 π 2 . (7.24) It is worth mentioning, as expected, that these are exactly the same choices that renormalize the 2-point function JJ and 3-point function T T J as well as all the other n-point functions involving two conserved currents. Furthermore, the counterterm contribution satisfies the constraints It is worth mentioning that this counterterm renormalizes the 2-point function JJ , perturbatively expressed as with B 0 (p 2 4 ) denoting the scalar (bubble) 2-point function, where the divergent part is extracted in DR as Coming to the the trace Ward identities, we also have At this point, having identified the anomalous conservation and trace WIs satisfied by the correlator, its renormalized expression is given by where the anomaly contribution is given by These direct computations are performed with no reference to the reconstruction procedure into transverse/traceless, trace and longitudinal sectors introduced in the previous sections, that allows the identification of a minimal number of form factors.

The reconstruction of the renormalized correlator from the transverse-traceless sector
Once we have removed the divergences from the transverse traceless part, we are able to reconstruct the entire correlator. The transverse traceless part, after the renormalization, acquires an anomalous contributions, with an anomalous dilatation WI satisfied by the corresponding form factors. The longitudinal part, instead, is affected by the presence of the trace anomaly. In this section, we are going to discuss this point in more detail. The renormalized correlator is given by where the bare correlator and the counterterm are re-expressed in terms of a finite renormalized correlator not contributing to the trace Ward identity, and a second part which accounts for the trace anomaly. In order to show this, we consider the first term in the longitudinal part of the correlator in (4.11), for which the divergent contribution is given by By adding the counterterm part, we are able to renormalize this expression, and taking the limit d → 4 we find where the anomalous part is explicitly given by 35) and where the T JJ anom 3-point function is given in (7.31). We make this contribution explicit by using (7.31) to obtain where we have defined Then, the second term in (4.11) is obtained from T µ 1 ν 1 t µ 2 ν 2 loc J µ 3 J µ 4 (d=4) anom with the replacement (1 ↔ 2). Similarly, one writes the last term t loc t loc JJ anom , as presented in Appendix C, in order to write the entire anomalous contribution given as In this expression we have defined with the properties where i = j ∈ {1, 2}, and the 0-residue term, instead, is written explicitly as with the property These relations show that the anomaly part of the action satisfies the conservation Ward identity 45) and the anomalous trace Ward identity Therefore, the anomaly part of the correlator satisfies the conservation WI by itself. A direct, explicit computation of the diagrams allows to check these relations for d = 4. The explicit anomalous conformal Ward Identities satisfied by the single form factors of the decompositions will be discussed elsewhere.

4-point functions from anomaly induced actions
In this section we turn our attention towards a study of the same correlator using the formalism of the anomaly induced actions. Actions of these type are variational solutions of the anomaly constraint. We denote by S A [g] the functional identified by the integration of the trace anomaly This functional is obtained by a Weyl rescaling of the metric g µν (x) = e 2φ(x)ḡ µν (x), to give S A [g] = S[ḡ] + ∆ S[σ,ḡ], (8.2) such that its conformal variation is the anomaly. The explicit expression for ∆S can be obtained after the integration of (8.3) along a path After the integration ∆S is written as (8.5) up to terms which are σ independent, i.e. conformally invariant, and hence do not contribute to the variation (8.3). In (8.5) we have defined the fourth order operator ∆ 4 as In order to obtain a nonlocal form of the anomaly induced action, one can impose a condition χ(g) = 0, such that the relation χ(g e −2φ ) = 0 is satisfied. From this condition one solves for φ in terms of a function of the metric, mainly φ = Σ(g), such thatḡ Thus the conformal decomposition is expressed as whereS[g] and S A [g, Σ(g)] are its conformal-invariant and anomalous parts respectively. There are two distinct exactly solvable conformal choices in 4D, in which the gauge parameter Σ(g) can be calculated in a closed form, as a functional of g. They are discussed in [45]. One is due to Fradkin and Vilkovisky (FV) [46] χ F V (g) = R(g) (8.10) with where ( − R/6) −1 is the inverse of the operator − R/6, playing the role of the corresponding Faddeev-Popov operator. Another choice is due to Riegert [47] with for which with (∆ 4 ) −1 the inverse of the operator in (8.6).

Variation of the anomaly effective action
In general, the two choices of Σ should be equivalent, giving the same result when the action is varied with respect to the metric. In order to see if this is the case, we expand S A [g, Σ(g)] around flat space by the simultaneous expansions of the metric g µν , the gauge field A µ , and the local factor Σ The local effective action from (8.9) is then written at the first order in h µν as neglecting the terms that are pure gravitational. We are assuming that Σ (0) = 0 as we will prove it in the next sections for the two choices of the parameter Σ. The first order in the fluctuation h µν of the expansion is related to the structure of the T JJ anomaly contribution. In order to extract the contribution to the T T JJ we need the second order in h µν , given as

The Riegert Action
We consider first the Riegert choice, for which where we have used the relation on the Green function of the ∆ 4 . Expanding (8.17) and taking order by order in the power of the fluctuations we determine the relations Substituting this expression into (8.15) we obtain the contribution to the T JJ as while the second order term is expressed as 20) and by using the expansion of the Paneitz operator ∆ 4 and integrating by parts we obtain In the expression above, the three terms affected by the Green function of 2 identify logarithmic contributions which can be correctly defined only by the inclusion of a scale µ F , in order to define their Fourier transform to momentum space. Indeed they can be correctly transformed to momentum space only in the presence of such cutoff. Obviously, one would expect such cutoff to disappear from the final expression, in such a way that the correlator satisfies, in its anomaly part, the same WIs valid for the anomaly part of its perturbative realization.
In momentum space, we obtain where G(p) = c log(p 2 /µ 2 F ) as pointed out also in [48][49][50] and we have used the relations in Appendix F. One can easily figure out that the µ F dependence in the sum logarithmic correlators does not cancel. As we are going to discuss next, a similar problem emerges in the case of the FV gauge. This, as we are now going to clarify, indicates an inconsistency of the expansion around the flat spacetime limit. Indeed, a close look at (7.45), the hierarchical equation of the conservation WI, relates on the lhs a 4-point function which remains logarithmic even after an ordinary differentiation, to 3-point functions on the rhs which are not, and are correctly described by the two anomaly-induced actions that we have considered. In the case of the Riegert action, therefore, this shortcoming appears to be related to 1/ 2 terms. This could provide an indication that these types of action do not allow a consistent expansion starting from the quartic order.

Fradkin-Vilkovisky Gauge
We consider then the Fradkin Vilkoviski choice for which In order to obtain the first and second order contributions in h µν with this choice we need the relations of expansion of the Green function of the conformal Laplacian. These are obtained considering and expanding and collecting the terms order by order we obtain With this results we use (8.25) to get It is worth noticing that the two choices are equivalent at the first order, indeed and then the anomalous contribution to the three point function T JJ is the same by using one choice or the other. Indeed, we obtain showing the equivalence of the two choices at the first order. At the second order, by inserting (8.28) in the expansion (8.17), we have (8.31) noticing that at the second order the expansions with the two different choices of Σ give distinguished results, and they differ for terms like x + . . . (8.32) We compute now the contribution of the anomaly effecgive action with the Fradkin Vilkovisky gauge, to the T T JJ in momentum space. From (8.31) and by using the variations in Appendix F we have One can check by explicit computation in momentum space -in this case the expression is transformable by the Fourier integral -that the trace Ward identity is violated. Indeed we have This is, obviously, at variance with the trace Ward identity satisfied by the anomaly contribution coming from the perturbative expansion discussed in the previous sections. In the case of the FV action, such Ward identities are expected to be expressed only in terms of hierarchical equations involving only the 3-and 4-point functions identified by the same action. This result raises some important issues concerning the consistency of the flat spacetime limit of such actions. Notice that the FV action also introduces a conformal decomposition related to a logarithmic gauge choice. The corresponding expansion, in FV, is also essentially deprived of a physical scale, although it is possible to formulate an explicit expansion of such gauge conditions in terms of the dimensionless combination R −1 .

New proposal
In this section we propose the solution of the problem discussed so far. The result we are going to discuss comes from the explicit calculation of the anomalous part of the correlator T T JJ performed in Section 7.3. In particular we write the correct expansion at the second order in curvature of an effective action that reproduces (7.38). We find that at the second order in the metric fluctuation, the desired expansion of the anomaly effective action should be The contribution to the anomaly part of the T T JJ of this expansion in momentum space reads as Using the expression of the functional variations in momentum space given in Appendix F, we write where the first two terms are exactly those in (7.39) and (7.44), and the last term is a Weyl invariant term written as and it does not contribute to the trace anomaly part. In principle this Weyl invariant contribution can be removed by an inclusion of a Weyl invariant term in the effective action. It is worth mentioning that the expression derived satisfies correctly the conservation and anomalous trace Ward identities.

Conclusions
In this work, we have performed an explicit computation of the T T JJ correlator using a free field theory realization. We have identified the general structure of the corresponding form factors in its tensorial decomposition.
In particular, we have implemented a method that simplifies their number, exploiting the momentum dependent degeneracies in combination with the classification of their orbits. The approach we presented is general and will be applied to similar correlators in future works. We have used this correlator to investigate the structure of the anomalous CWIs satisfied by it and compared the results with the analogous prediction for its anomaly part, as identified by the anomaly-induced actions. Such actions are expected to reproduce the anomaly contribution to all orders in the external gravitational field. Indeed, this has been verified in the case of 3-point functions for two special choices of the conformal decomposition of the background metric that generate two different types of anomaly actions.
Indeed we have shown that, for 3-point functions, the two actions of Riegert and FV type are both in agreement with the perturbative analysis, once that we have isolated from the perturbative correlator its anomaly contribution.
As explained in the previous sections, this separation is uniquely defined since the anomaly part of a correlator is identified by the condition that 1) it reproduces the anomaly contribution in the hierarchy and 2) it is separately conserved. In other words, such part satisfies anomalous CWIs while the non anomalous part satisfies CWIs which are ordinary (non anomalous), and both parts are separately conserved.
As we move to 4-point functions, our analysis shows that the conservation WI for the anomaly part should be modified by an additional traceless contribution (that we have called the "0-residue" part). In other words, at the level of 4-point functions, the anomaly part of the correlator is not just identified by the sequence of pole-like contributions, which are part of the trace WIs, but also by a traceless part. This allows, again, to separate the correlator into two parts, separately conserved and satisfying, again separately, anomalous CWIs and ordinary CWIs, respectively, as for 3-point functions. These analyses demonstrate the important role of free field theory realizations in extracting such information, which is not a priori predictable from general considerations in this class of theories. This point has been first noticed in a previous analysis of the anomaly contribution to the T T T T (4T ) [3].
In the case of the 4T correlator, that we hope to investigate in the future by similar methods, such (0-residue) part could be identified by the expansion of the counterterms 1 V E and 1 V C 2 , without the need to determine the entire structure of the correlator. Indeed, one only needs to perform a consistent decomposition of the counterterms up to O(1) in to achieve the goal. We recall that the anomaly, in DR, comes for the expansion of V E/C 2 /F 2 up to O( ), which in flat space is polynomial in the momenta. The application of the transverse-traceless/longitudinal/trace decomposition, introduced in [7], has allowed extracting such 0-residue term from the finite part of the counterterms, indicating that such terms are essential in the definition of an anomaly action defined directly from the perturbative expansion. Therefore, the anomalous CWIs and the conservation WI are both satisfied only if we add to the pole-term contributions such 0-residue term. The perturbative analysis of the T T JJ that we have discussed is similar to the case of the 4T discussed in previous work, and exhibits the same features.
On the other end, our analysis shows that the anomaly-induced actions that we have investigated fail -at the level of 4-point functions -to generate conserved anomaly parts, while they succeed in 3-point functions. At the same time, as we have shown, they do not satisfy the hierarchical CWIs that we expect. The result is puzzling -but rather interesting -since it may bring under even closer scrutiny this class of actions, originally introduced as formal solutions of a variational problem. We also remark that conservation WIs are associated with the diffeomorphism invariance, and one indeed expects that the conservation of the stress energy tensor should hold. Indeed it does, at least for the correlators of rank 2 and 3.

Possible resolutions
Our conclusions, given these findings, are open-ended and call for further investigations of such correlation functions. One crucial issue that needs to be addressed is if such actions can be consistently expanded around a flat spacetime -in the absence of any physical scale in their expressions -beyond a specific order, without encountering the puzzling behaviour that we have identified in our analysis. The need to introduce an extra scale µ F in the expression of the T T JJ vertex using Riegert's conformal decomposition, is an indication that the action cannot be correctly Fourier transformed to momentum space. This is in clear contradiction with the result we have obtained for the anomaly part of the perturbative T T JJ and corresponding WIs, which are, obviously, free of any logarithmic term. In Riegert's action, this point is, in a way, expected, given that the ∆ 4 operator is quartic and it has been noticed before [5].
It is nevertheless remarkable that the action successfully predicts the behaviour of the T T T , as shown in [2]. On the other hand, the FV action does not suffer from such shortcomings since it can be transformed in momentum space. However, the gauge choice (Σ F V ) used for its definition is formally very involved, being defined as a logarithmic expansion of a curvature dependent Green function. Even though no scale is present in the logarithm, one may ask if the expansion is justified in the flat spacetime limit, and R −1 is a reasonable variable that can appear in an expansion. The issues that we have identified could be related to such intrinsic limitations. The answer to this question may be found by extending these types of analysis to a curved space, to begin with, such as to a maximally symmetric space or to other spaces where dimensionful constants are naturally present. Finally, one possibility is to proceed with a modification of such anomaly induced action by the inclusion of Weyl invariant terms. These extra terms can be identified, by investigating more closely the mismatch between the perturbative analysis and the predictions of such anomaly-induced actions, as done in this work for the T T JJ.
In all these analyses, it is obvious that the safest way to deal with these problems is to proceed with perturbative tests, that provide a safe reference for any further investigation of these issues. In principle, the strategy we have presented can be extended any n-point function. These important aspects are currently being investigated.

A.1 Diffeomorphism invariance
In this section, we derive the conservation and trace Ward identities for the T JJ and T T JJ. We start from the invariance under diffeomorphism, for which the energy momentum tensor has to satisfy that can be re-written as 2 ∂ µ 1 δS δg µ 1 ν 1 Taking functional variations with respect to the gauge fields we find where we have discarded the terms that in the limit A → 0 does not contribute. Taking one more functional derivative with respect to the metric and neglecting terms that vanish in the limit g → δ we have Then by using the explicit expressions of the functional derivatives written as 3) to obtain the conservation Ward identities for the T JJ as and from (A.4) the conservation Ward identities for the T T JJ reads as In momentum space, these conservation Ward identities are written as

A.2 Gauge invariance
The invariance under gauge transformation implies the conservation of the current By taking functional derivatives with respect to the metric and the gauge fields and considering the flat spacetime limit, we find the conservation Ward Identities for the 3and 4point functions that in momentum space become and similar equations for the contraction of p 4µ 4 .

A.3 Weyl invariance
The invariance under Weyl transformation implies that the energy momentum tensor has zero trace, mainly in general d dimensions. Taking functional derivatives of the previous equation we have the trace Ward identities In d = 4, due to the quantum effect, the trace of the energy momentum tensor acquires a non-zero trace contribution leading to the anomalous trace Ward identities for the correlation functions where the functional derivatives of the field strength F 2 with respect tot he gauge field are expressed in (F.7) and (F.8).
and for the δpppp sector we have and finally the last sector pppppp is identified, as in the general case, by the representatives B.3 The case of d = 3 In this section we list the representatives of the orbits for each sector of tensor structures used to construct the transverse traceless part when d = 3.

D Metric variations involving the Vielbein
We present in this section all the useful relations related to the metric variation of the vielbein. In particular we have that the variation with respect to the metric tensor can be written as and these relations are in accordance with the well known variation of the curved metric Other useful relations in order to get the vertices are written as δ (V V µ a ) δ g µ 1 ν 1 = V 2 g µ 1 ν 1 V µ a − g µ(µ 1 V ν 1 ) a (D.5) δ 2 (V V µ a ) δ g µ 1 ν 1 δg µ 2 ν 2 = V 2 −g µ 1 (µ 2 g ν 2 )ν 1 V µ a + g µ(µ 2 g ν 2 )(µ 1 V ν 1 ) a + V 4 g µ 1 ν 1 g µ 2 ν 2 V µ a − g µ(µ 2 V ν 2 ) a − V 4 g µ(µ 1 V ν 1 ) a g µ 2 ν 2 + V 4 g µ(µ 1 g ν 1 )(µ 2 V ν 2 ) a (D. 6) and we observe that δ 2 (V V µ a ) δ g µ 1 ν 1 δg µ 2 ν 2 = δ 2 (V V µ a ) δ g µ 2 ν 2 δg µ 1 ν 1 (D.7) We can obtain the metric variation of the spin connection starting from its definition Then its variation variation under a metric variation is given as (D.10) Then the variation with respect to the metric is obtained as It is worth mentioning that, since the spin connection always appears contracted with the antisymmetric tensor Σ ab , we can omit the antisymmetrization on the indices [a, b].