Parity-Violating CFT and the Gravitational Chiral Anomaly

We illustrate how the Conformal Ward Identities (CWI) and the gravitational chiral anomaly completely determine the structure of the $\langle TTJ_{5}\rangle$ (graviton-graviton-chiral gauge current) correlator in momentum space. This analysis extends our previous results on the anomaly vertices $\langle AVV\rangle$ and $\langle AAA\rangle$, as well as the $\langle TJJ\rangle$ parity-odd conformal anomaly vertex in general CFTs. The $\langle TTJ_{5}\rangle$ plays a fundamental role in the analysis of the conformal backreaction in early universe cosmology, affecting the particle content and the evolution of the primordial plasma. Our approach is nonperturbative and not Lagrangian-based, requiring the inclusion of a single anomaly pole in the solution of the anomaly constraint. The pole and its residue, along with the CWIs, determine the entire correlator in all of its sectors (longitudinal/transverse), all of which are proportional to the same anomaly coefficient. The method does not rely on a specific expression of the CP-odd anomalous current, which in free field theory can be represented either by a bilinear fermion current or by a gauge-dependent Chern-Simons current; it relies solely on the symmetry constraints. We compute the correlator perturbatively at one-loop in free field theory and verify its exact agreement with the non-perturbative result. A comparison with the perturbative analysis confirms the presence of a sum rule satisfied by the correlator, similar to the parity-even $\langle TJJ\rangle$ and the chiral $\langle AVV\rangle$.


Introduction
The original approach to identifying correlation functions in conformal field theories (CFTs) has traditionally been formulated using coordinate space methods, both for scalar and tensor correlators.In the presence of anomalies, the solutions of the corresponding conformal Ward identities (CWIs) have been obtained by partitioning the domain of definition of each correlator into nonlocal and contact contributions.The equations are initially solved for the regions in which the external coordinate points of the correlators are all noncoincident.Anomalous corrections, which arise when all the points coincide, are manually added by including additional local terms with support defined by products of delta functions.This methodology was pioneered in groundbreaking works [1,2], and it was successfully applied to correlators containing the stress-energy tensor (T ) and conserved vector currents (J), specifically the ⟨T T T ⟩ and the ⟨T JJ⟩ correlators, respectively.On the other hand, investigations of the conformal constraints in momentum space, in the presence of conformal anomalies, are more recent.This approach has been explored in several works [3,4,5,6,7,8], using a general formulation, and it has been further examined in perturbation theory [9,10,11] for correlation functions such as the ⟨T JJ⟩ and the ⟨T T T ⟩, using free field theory realizations.The analysis of 4-point functions in both generic CFTs and in free field realizations has been discussed in [12,13].These analyses have predominantly focused on the parity-even sector, including the contribution of the conformal anomaly.In contrast, investigations into anomaly-free correlators of odd parity have only recently emerged [14,15,16].Given the intricate nature of chiral and conformal anomalies, which are related to contact interactions, the coordinate approach becomes unwieldy, and the hierarchical character of the CWIs certainly becomes rather involved.Consequently, shifting to momentum space offers advantages due to its connection with ordinary off-shell scattering amplitudes.
1.1 The ⟨T T J 5 ⟩ from parity-odd CFT Gravitational anomalies generated by spin 1/2 and spin 3/2 particles have been extensively studied in several works since the 70's, due to their connection with ordinary gauge theories [17,18], supergravity [19] and self-dual antisymmetric fields in string theory [20], just to mention a few (see also [21] for a recent study on the properties of chiral anomalies in the context of black holes).The gravitational anomaly R R can appear in different settings.The nonperturbative approach developed in this paper is general and adaptable to many contexts, yet the anomaly's impact can vary from benign to dangerous, based on the circumstances.Consider, for example, a scenario involving a Dirac fermion interacting with gravity and a vector potential V µ .Kimura, Delbourgo and Salam were the first to compute the anomaly in this case, observed in the divergence of J 5 [17,18].This specific anomaly poses no threat and may be of interest in phenomenology.For convenience, we can also introduce an axial-vector field A µ , which couples to J 5 , but only as an external source since an anomalous gauge symmetry for A µ would spoil unitarity and renormalizability.Another instance involves a chiral model incorporating a Weyl fermion ψ L/R interacting with gravity and a gauge field.In this case, the anomaly emerges in the divergence of J L/R , potentially endangering unitarity and renormalization, unless it is canceled [20].See [22] for a detailed account on the types of chiral anomalies and their relation to diffeomorphism invariance.In general, in perturbation theory, the evaluation of a chiral trace of Dirac matrices hinges on the choice of a specific regularization and the related treatment of the antisymmetric ε tensor in the loop.In the case of the Breitenlohner-Maison-'t Hooft-Veltman scheme [23], for example, the anomaly of parity-odd correlators is present only on the Ward identity of the chiral current, while the energy-momentum tensor and the vector currents are conserved.In other regularizations, one can potentially find a violation of the latter as well.The correlator under scrutiny in this work is the ⟨T T J 5 ⟩, reinvestigated using CFT in momentum space (see [24] for a review).We will utilize a formalism developed for curved spacetime, from which the flat spacetime CWIs will be consistently derived in d = 4, as constraints from special background metrics [25].This correlator involves two stress-energy tensors and one parity-odd current, denoted as J 5 .A study of this correlator was previously discussed in [26] using coordinate space methods.In the Standard Model, when J 5 is the non-Abelian SU (2) gauge current or the hypercharge gauge current, this anomaly cancels out by summing over the chiral spectrum of each fermion generation.This feature is usually interpreted as an indication of the compatibility of the Standard Model when coupled to a gravitational background, providing an essential constraint on its possible extensions.The correlator plays a crucial role in mediating anomalies of global currents associated with baryon (B) and lepton (L) numbers in the presence of gravity.In condensed matter theory, correlators affected both by chiral and conformal anomalies, as well as by discrete anomalies, play an important role in the context of topological materials [27,28,29,30,31].In our analysis, we demonstrate how investigating CFT in momentum space allows us to independently reproduce previous results found in coordinate space [26].The solution is uniquely constructed by assuming the exchange of a single anomaly pole in the longitudinal sector of this correlator when we proceed with its sector decomposition.We will show that the momentum space solution, derived from the CFT constraints, is unique and depends on a single constant: the anomaly coefficient at the pole.This result appears to be a common feature in correlation functions that are finite and affected by parity-odd anomalies, complementing our previous analysis of similar correlation functions such as the ⟨JJJ 5 ⟩ (or ⟨AV V ⟩ chiral anomaly vertex) and the parity-odd ⟨T JJ⟩.However, there are also some differences between the ⟨AV V ⟩ and ⟨T T J 5 ⟩ cases, which are affected by a chiral anomaly, and the ⟨T JJ⟩ odd , when this correlator is assumed to develop a parity-odd trace anomaly.In the ⟨AV V ⟩ and the ⟨T T J 5 ⟩, both the longitudinal and transverse sectors are nonzero and completely determined by the anomaly pole, initially introduced in the longitudinal sector as a solution of the anomaly constraint [32].Instead, in the ⟨T JJ⟩ odd , only the longitudinal sector survives after imposing the conformal constraints together with the (chiral) trace anomaly [33], while the remaining sectors vanish.

Fermionic and Chern-Simons currents
In free field theory, two realizations of currents have been discussed for this correlator: the bilinear (axial-vector) fermion current J 5f and the bilinear gauge-dependent Chern-Simons (CS) current (J 5 ≡ J CS ) [34,35].We recall that in previous analysis it has been shown that both currents of the form or of the Chern-Simons form could be considered in a perturbative realization of the same correlator and generate a gravitational anomaly.Notice that this second version of the current can be incorporated into an ordinary partition function -in an ordinary Lagrangian realization by a path integral -only in the presence of a coupling to an axial-vector gauge field (A λ ) via an interaction of the form The term, usually denoted as A V ∧ F V , is the Abelian Chern-Simons form that allows to move the anomaly from one vertex to another in the usual ⟨AV V ⟩ diagram.Details on these point can be found in [36,37].Notice that both currents satisfy the parity-odd constraint given in (14).J CS is responsible for mediating the gravitational chiral anomaly with spin-1 virtual photons in the loops, resulting in a difference between their two circular modes and inducing an optical helicity.This interaction is relevant in early universe cosmology and has an impact on the polarization of the Cosmic Microwave Background (CMB) [38].
In this case, the classical symmetry to be violated is the discrete duality invariance (E → B, B → −E) of the Maxwell equations in the vacuum (see [39,40]).The ⟨T T J 5 ⟩ correlator induces similar effects on gravitational waves [41,42].Spectral asymmetries induced by chiral anomalies, particularly the ordinary chiral anomaly (the F F ∼ E • B term), have been investigated for their impact on the evolution of the primordial plasma, affecting the magneto-hydrodynamical (MHD) equations and the generation of cosmological magnetic fields [43,44,45].
As previously mentioned, our method exclusively exploits the correlator's symmetries to identify its structure, which remains identical for a generic parity-odd J 5 .In both cases (J 5f and J CS ), the solution is entirely centered around the anomaly pole, serving as a pivot for the complete reconstruction of the corresponding correlators.Both realizations of the ⟨T T J 5 ⟩ correlator -using a J CS current or a J 5f current -have been shown in [34,35] to reduce to the exchange of an anomaly pole for on-shell gluons and photons for the unique form factors present in the diagrams.In these works, the authors introduced a mass deformation of the propagators in the loops and showed the emergence of the pole as the mass was sent to zero.The method relies on the spectral density of the amplitude and it has been used also more recently in [46] and [47], in studies of the parity-even ⟨T JJ⟩ and in supersymmetric variants.We comment on this point in Section 9 and illustrate, by a simple computation, that the spectral densities of the only surviving form factors in the on-shell ⟨T T J 5 ⟩, with J 5 ≡ J 5f and J 5 ≡ J CS , satisfy two (mass-independent) sum rules.

Organization of the work
The outline of the paper is as follows.
In Section 2 we review some of the feature of the perturbative approach.We examine the link between anomalies and the presence of poles in the expression of correlators in momentum space.
In Section 3, we briefly comment on the methodology followed in the solution of the 4d CWIs in two previous analyses by us [32,33] involving anomalous parity-odd correlators, which may help clarify some of the technical points contained in this work.Then, in Section 4, we examine the constraints following from diffeomorphism, gauge and Weyl invariance.In particular, we express the conformal constraints on the ⟨T T J 5 ⟩ as 4d differential equations first in coordinates and then in momentum space.The two following sections then discuss the general decomposition of the correlator, following the methods of [4], extended to the parity-odd case, and the solution of the conformal constraints.We present the general expression of the conformal ⟨T T J 5 ⟩ correlator in momentum space.In Section 7, we perform a perturbative analysis of the correlator and in Section 8 we verify that the conformal solution and the perturbative one coincide.Then, in Section 9 we show that, similarly to previous dispersive analysis of the anomalous form factors for the ⟨T JJ⟩ and ⟨AV V ⟩ diagrams, the spectral density of the anomalous form factor of the ⟨T T J 5 ⟩ satisfies a sum rule.We summarize our findings in Section 10 and discuss the non-renormalizability of the ⟨AV V ⟩ and ⟨T T J 5 ⟩ in Section 11.We leave to the Appendix a discussion of some technical points concerning 3K integrals, the use of the Schouten relations and the identities we have used to identify the correlator by the perturbative and non-perturbative methods.
It is worth noting that the analysis of the conformal constraints and the reconstruction method of [4], here implemented in the parity-odd case, allows us to express the correlator in terms of a minimal number of form factors.For example, in the case of the parity-even ⟨T JJ⟩, the reduction in their numbers has been from thirteen down to three [4][10] by the inclusion of the conformal constraints and a special choice of the parametrization of the tensor structures symmetric in the external momenta.A similar analysis has been performed for 4-point functions in the parity-even ⟨T T JJ⟩ [13] and ⟨T OOO⟩ [12], where O are identical conformal primary scalars.

The nonperturbative approach
We will investigate the structure of the full (uncontracted) vertices, in contrast to the majority of previous literature that has focused solely on computing the anomaly, namely the WI.
As demonstrated in previous works through perturbative analyses of correlators such as the ⟨JJJ 5 ⟩ and the parity-even ⟨T JJ⟩ [46,48], or the superconformal anomaly multiplet [47], the anomaly in momentum space is associated with the exchange of an anomaly pole [49].While the coordinate space approach is valuable, it has limitations in revealing the underlying dynamical source of the anomaly.This aspect becomes considerably clearer in momentum space when conducting a dispersive analysis of the anomaly form factor.
In the case of the ⟨T T J 5 ⟩, previous analyses, conducted for J CS , have identified the presence of such pole in the correlator [34].This analysis was based on perturbation theory for on-shell gravitons.
We will provide further comments on these previous findings in a following section.It has been established, in a somewhat general context, that various types of anomalies (chiral, conformal, supersymmetric, etc.) are invariably characterized by the presence of a form factor with a pole, which multiplies a tensor structure responsible for generating the anomaly.The nonlocality of the anomaly is, therefore, summarized, at least in the context of flat spacetime, by an effective action of the form [46][48].
for an anomalous ⟨J 5 JJ⟩ (axial-vector/vector/vector) vertex, with A µ denoting an external axial vector and V µ a vector gauge field.A similar nonlocal action can be written down for the parity-even ⟨T JJ⟩ vertex, with one stress-energy tensor T and two conserved vector currents J [46,50].R (1) is the linearized Ricci scalar.The ellipsies in the expres-sions above indicate terms which do not contribute to the anomaly, either chiral (4) or conformal (5).If we identify the ⟨AV V ⟩ vertex in momentum space as the contribution isolated in ( 4) is obtained from the solution of the constraint in the form with the ellipsis referring to to the transverse components.The 1/p 2 3 contribution is the anomaly pole.In a similar fashion, in the ⟨T JJ⟩, the momentum space analysis reveals that the correlator is decomposed in the form [46,48] where b 4 is the anomalous coefficient in eq. ( 12) and is given by the Fourier transform of the anomaly functional (F F ) differentiated with respect to the external gauge field It is evident that, based on this information, one anticipates that each anomalous vertex should exhibit a nonlocal interaction in flat spacetime, mirroring the two aforementioned instances.What presents a greater challenge is the determination of whether the inclusion of a pole as a solution to the anomalous WI, in conjunction with the conformal constraints, suffices to ascertain the complete expression of the correlators.In other words, concerning the parity-odd correlators ⟨T JJ⟩, ⟨AV V ⟩, and ⟨T T J 5 ⟩, it is conceivable that the 3-point functions can be wholly reconstructed from the residue at the pole, which is defined by the anomaly coefficient.
It is worth noting that, within our methodology, the pole is not directly associated with the current but with the entire correlator itself.Hence, in the context of the ⟨T T J 5 ⟩ correlator under investigation in this study, the specific form of the parity-odd operator current that satisfies the anomaly constraint does not play an essential role.

Parity-odd terms
A common track of parity-odd correlation functions is the presence of anomaly constraints, either in the form of chiral trace anomalies, or just of chiral anomalies, inducing CP violation.In the case of chirally-odd trace anomalies, which have been at the center of several recent debates, the anomaly functional, constraining the form of the correlator, is related to CP violating terms beside the usual parity-even ones in the form where F µν can be either an axial or vector gauge field.Parity-odd terms corresponding to the constant f 1 and f 2 have been conjectured long ago by Capper and Duff [51,52] on dimensional grounds and by requiring covariance.From a cohomological point of view, such term are consistent and cannot be discarded.In previous work we have shown that the CWIs severely constrain the structure of parity-odd correlators [33].For example, a correlator such as the (rank-4) ⟨T JJ⟩ odd , if we resort to a longitudinal/transverse-traceless/trace decomposition of its tensorial structure, is constrained only to contain a nonvanishing trace sector, with the remaining sectors being zero.We will illustrate a similar decomposition in the case of the ⟨T T J 5 ⟩, that we are going to investigate in the next sections.
A similar analysis has been performed in the case of the ⟨AV V ⟩ diagram [32], decomposed into a transverse and a longitudinal sectors.We have shown that the entire chiral anomaly vertex can be derived from CFT by solving the anomaly constraints, with the anomaly attributed to the axial-vector current The reconstruction of the correlator is performed using as a pivot the anomaly pole present in the decomposition of this correlator in its anomaly sector.In the case of the ⟨JJJ 5 ⟩ or of the ⟨J 5 J 5 J 5 ⟩, we satisfy the anomaly constraint with one or three anomaly poles respectively, and all the remaining sectors are fixed by this choice.The general character of this reconstruction procedure, based on the inclusion of the anomaly constraint, will be proven also in the case of the ⟨T T J 5 ⟩.In all these three cases, our analysis shows that the anomaly phenomenon, at least in the parity odd case, is entirely associated with the presence of an anomaly pole.Once the coefficient in front of the anomaly is determined, then the entire correlator is fixed, if we impose the conformal symmetry.In the ⟨T T J 5 ⟩, if we include in the background both a gauge field and a general metric background, the anomalous WI that we will be using for the definition of the correlator in CFT is given by the equation that defines a boundary condition for the CWIs.These results can also be extended to the non-Abelian case.For a general chiral current J µ i , we can write where we have introduced the anomaly tensors constructed with the non-Abelian generators of the theory.In the case of the D i 's, for example, in the Standard Model, where the symmetry is SU (3) × SU (2) × U (1) Y , only the hypercharge (U (1) Y ) contribution ⟨T T J Y ⟩ is taken onto account, since the SU (2) and SU (3) generators are traceless.Both the chiral (F F ) and gravitational (R R) anomalies cancel once we sum over each generation of chiral fermions [53].The cancellation of the gravitational anomaly in the Standard Model can be interpreted in two possible ways.On one hand, it shows the consistency of the coupling of the Standard Model to gravity, since the gauge currents are conserved in a gravitational background.
On the other hand, the stress-energy tensor is just another operator of the Standard Model and the conservation of the currents is required for the analysis of the mixing of such operator with the gauge currents at perturbative level.

Ward identities
The correlator is constrained by diffeomorphism invariance, gauge invariance and the CWIs, derived from Weyl invariance.For this purpose, it is convenient to obtain the identities directly from the functional integral, by requiring the invariance of the effective action S in the Euclidean space under the corresponding symmetries.The integration runs over all the matter/radiation fields, here denoted as Φ.The equations can also define the constraints for any non-Lagrangian CFT.
We couple the external gauge field source A µ to the current J 5 .We define the quantum averages of the energy-momentum tensor and of the current in terms of the generating functional of the theory from which we derive the correlator of interest

• Diffeomorphism invariance
We start from diffeomorphism invariance.Under a diffeomorphism the fields transform with a Lie derivative The WIs follow from the requirement that the generating functional S is invariant under these transformations obtaining In order to find a WI for the correlator ⟨T T J 5 ⟩ we need to apply to this equation the functional derivatives δ δg δA .Due to the vanishing of all 1-point and 2-point functions in 4d in the limit g µν → δ µν and A µ → 0, only the first term in the eq.( 22) survives.Going to momentum space, this procedure generates the equation • Gauge invariance We proceed in a similar manner for gauge invariance.The action of a gauge transformation, with parameter α(x) on the fields, gives the infinitesimal variations The requirement that the generating functional S is invariant under these transformations leads to the conservation of the current J α 5 .If we allow an anomaly from the path integral measure we obtain the relation We now apply two functional derivatives with respect to the metric to such equation, and perform the limit g µν → δ µν and A µ → 0 as above.After going to the momentum space, we derive the relation This constraint will be satisfied by the inclusion of an anomaly pole in the correlator.

• Weyl invariance
By requiring the invariance of the partition function under the conformal transformations in flat space, we identify five extra constraints, beside those related to diffeomorphism invariance, corresponding to the special conformal transformations and the dilatations.Taken together, diffeomorphism plus Weyl invariance in curved spacetime determine in any local free-falling frame associated with the metric g µν , the symmetry constraints of the conformal group on the correlator.The derivation of these constraints, from a curved spacetime perspective, requires a background metric that allows conformal Killing vectors.This imposes a formal restriction on the class of metric backgrounds with respect to which the functional variations are performed.In other words, this approach allows to patch together the constraints in each local frame.If the classical action is Weyl invariant, a metric that allows conformal Killing vectors requires that σ(x) is at most quadratic in the local coordinates, and is expressed in terms of 15 parameters, which are the parameter of the conformal group.We try to clarify this point, that has been discussed in [25] in the case of the ⟨T T T ⟩ correlator, extending it to the ⟨T T J 5 ⟩, with its specific expression of the anomaly.The action of a Weyl transformation with a dilaton σ(x) on the fields, acting as a parameter is If the metric is selected in such a way to allow conformal Killing vectors, then the diffeomorphisms x µ → x ′µ = x µ + K µ (x) induce a simple rescaling of the infinitesimal distance, under a local rescaling with σ(x) This require that σ(x) and the same vectors are related At this stage, from the action we define the conformal currents J K , with their quantum averages given by that differentiated give At this stage we can use both the property of the background metric ( 29) and resort to the definition of the stress energy tensor to rewrite the equation into the form where The conservation of the conformal currents requires that δ δσ S 0 = 0, (35) plus the ordinary diffeomorphism invariance, that guarantees the vanishing of the second addend in (32) At quantum level (35) may be affected by a conformal anomaly with where A is the anomaly functional.The conservation of J K is violated in the case of such conformal anomaly.
The invariance of the generating functional S under the transformations ( 27) leads to the tracelessness of the energy-momentum tensor.However, in general we need to consider anomalous terms coming from the path integral measure After applying the functional derivatives δ δg δ δA and performing the limit g µν → δ µν and A µ → 0, the anomalous terms do not survive.Going to momentum space, we obtain the constraint which is a nonanomalous trace WI.We now consider the conformal transformations.The conformal Killing vectors for the dilatations are while for the special conformal transformation they are given by As discussed in [25], the derivation of the CWIs requires a rank-4 correlator and can be directly formulated in the local frame.In this case, we focus on the 4-point function ⟨T T T J 5 ⟩ and consider the divergence condition ) Recalling the conformal Killing vector equation ( 29), we can then derive the equation On the right-hand side of the last equation we have the trace and the divergence of a four-point correlator function.We can use the anomalous trace equation and the conservation of the energymomentum tensor in order to rewrite such terms.We will show this in the following.We first focus on the dilatations.The Killing vectors in this case are given by (23).The invariance under diffeomorphism of the partition function (22), differentiated and in the flat limit gives where δ xy is the dirac delta function δ 4 (x − y) and all the derivatives are with respect to the x variable.Similarly, we differentiate the anomalous trace equation (22) obtaining in the flat limit Inserting the equations ( 23), ( 30) and ( 32) into (28) and integrating by parts we obtain the ordinary dilatation Ward identity (WI) that in momentum space takes the form If, instead, we consider the special conformal transformations, the Killing vectors are given in (24).
Proceeding in a similar manner we arrive at the expression that in momentum space takes the form At this stage we are ready to proceed with the decomposition of the correlator into all of its sectors and derive the scalar equations for its reconstruction [4].We will first proceed with a parametrizaton of the form factors and tensors structures of the transverse-traceless sector.We introduce a form factor in the longitudinal part, in the form of an anomaly pole, and proceed with a complete determination of the entire correlation function by solving the equations of all the remaining sectors.We follow the steps introduced in [4], extended to the parity-odd case, and split the equations into primary and secondary CWIs.The solution, as we are going to show, will coincide with the perturbative one and will depend on a single constant, the coefficient of the anomaly.The off-shell parametrization of the vertex that results from this construction is quite economical, and is expressed in terms of only two form factors in the transverse traceless sector, plus the anomaly form factor that takes the form of a 1/p 2 3 anomaly pole.The anomaly, in this formulation, is the residue at the pole.

Decomposition of the correlator
In this section we find the most general expression of the ⟨T T J 5 ⟩ correlator, satisfying the anomalous conservation WI and trace WI.The analysis is performed by applying the L/T decomposition to the correlator.We focus on a parity odd four-dimensional correlator, therefore its tensorial structure will involve the antisymmetric tensor ε µνρσ .We start by decomposing the energy-momentum tensor T µν and the current J µ 5 in terms of their transverse-traceless part and longitudinal ones (also called "local") where having introduced the transverse-traceless (Π), transverse (π) and longitudinal (Σ) projectors, given respectively by Such decomposition allows to split our correlation function into the following terms Using the conservation and trace WIs derived in the previous section, it is then possible to completely fix all the longitudinal parts, i.e. the terms containing at least one t µν loc or j µ 5 loc .We start by considering the non-anomalous equations Thanks to these WIs, we can eliminate most of terms on the right-hand side of equation ( 56), ending up only with two terms The remaining local term is then fixed by the anomalous WI of J 5 .First, we construct the most general expression in terms of tensorial structures and form factors where, due to the Bose symmetry, both F 1 and F 2 are symmetric under the exchange (p 1 ↔ p 2 ).Then, recalling the definition of j 5 loc and the anomalous WI we can write (61) One can show that this formula coincides with eq. ( 59) after contracting the projectors' indices and fixing the form factors in the following way Therefore, all the local terms of the ⟨T T J 5 ⟩ are fixed.The only remaining term to be studied in order to reconstruct the entire correlator is the transverse-traceless part ⟨t µ 1 ν 1 t µ 2 ν 2 j µ 3 5 ⟩.Its explicit form is given by where X α 1 β 1 α 2 β 2 α 3 is a general rank five tensor built by products of metric tensors, momenta and the Levi-Civita symbol with the appropriate choice of indices.Indeed, as a consequence of the projectors in (63), X α 1 β 1 α 2 β 2 α 3 can not be constructed by using g α i β i , nor by p i α i with i = {1, 2, 3}.We also must keep in mind that, due to symmetries of the correlator, form factors associated with structures linked by a (1 ↔ 2) transformation (the gravitons exchange) are dependent.Then, the transverse-traceless part can be written as where A 3 and A 4 are antisymmetric under the exchange (p 1 ↔ p 2 ) and we have made a choice on which independent momenta to consider for each index Since we are working in d = 4 the form factors in eq. ( 64) are not all independent and the decomposition is not minimal.Indeed, one needs to consider the following class of tensor identities If we set α = β 1 or α = β 2 and apply the projectors, we have according to which we can rewrite the tensorial structures multiplying A 7 in terms of the others.
If we instead contract the identity (66) with p 1α and p 2α , we arrive to according to which we can rewrite the form factors A 5 and A 6 in terms of the first four.We conclude that the general structure of the transverse-traceless part is given by where we have redefined the form factors A 1 , . . ., A 4 .Once again, A 3 and A 4 are antisymmetric under the exchange (p 1 ↔ p 2 ).
6 The conformal analysis of the ⟨T T J 5 ⟩ In the previous section we have seen that the conservation and trace WIs fix the longitudinal part of the correlator.In this section we examine the conformal constraints on the ⟨T T J 5 ⟩, following closely the methodology adopted in [4].We will see that the transverse-traceless part of the correlator is completely determined by conformal invariance together with the R R part of the boundary condition coming from the anomaly relation (14), corresponding to the anomalous coefficient a 2 .

Dilatation Ward identities
The invariance of the correlator under dilatation is reflected in the equation (47).Due to this constraint, the transverse-traceless part of the correlator has to satisfy the equation By using the chain rule to express the derivatives respect to 4-vectors in term of the invariants p i = | p 2 i |, we rewrite (70) as a constraint on the form factors with N j the number of momenta that the form factors multiply in the decomposition of eq. ( 69)

Special conformal Ward identities
The invariance of the correlator with respect to the special conformal transformations is encoded in the following equation (74) The special conformal operator K κ acts as an endomorphism on the transverse-traceless sector of the entire correlator.Therefore we can perform a transverse-traceless projection on all the indices in order to identify a set of partial differential equations splitting the correlator into its transverse and longitudinal parts.The action of the special conformal operator K κ on the longitudinal part of the correlator is given by Using the eq.( 60) together with the Schouten identities mentioned in the Appendix B, we can write Using the Schouten identities reported in Appendix B, we can then decompose the action of the special conformal operator on the entire correlator in the following minimal expression where the coefficients C ij depend on the gravitational anomalous coefficient a 2 , the form factors A i and their derivatives with respect to the momenta.Due to the independence of the tensorial structures listed in the equation above, all the coefficients C ij need to vanish.In particular the primary equations are They correspond to second order differential equations.The secondary equations are instead given by 0 = and they are differential equations of the first order.

Solving the CWIs
The most general solution of the CWIs of the ⟨T T J 5 ⟩ can be written in terms of integrals involving a product of three Bessel functions, namely 3K integrals [4].For a detailed review on the properties of such integrals, see also [54].We recall the definition of the general 3K integral where K ν is a modified Bessel function of the second kind We will also use the reduced version of the 3K integral defined as where we introduced the condensed notation {k j } = {k 1 , k 2 , k 3 }.The 3K integrals satisfy an equation analogous to the dilatation equation with scaling degree [4] deg where From this analysis, it is simple to relate the form factors to the 3K integrals.Indeed, the dilatation WI tells us that the form factors A i can to be written as a combination of integrals of the following type where N i is the number of momenta that the form factors multiplies in the decomposition (69).The special CWIs fix the remaining indices k 1 , k 2 and k 3 .We start by considering the explicit form of the primary equations (79) involving the form factor where we have defined Recalling the following property of the 3K integrals we can write the most general solution of the primary equations as where ζ 1 is an arbitrary constant.Note that this solution is symmetric under the exchange of momenta p 1 ↔ p 2 .Indeed, from the definition of the 3K integral, it follows that for any permutation σ of the set {1, 2, 3} we have However, due to the Bose symmetry, the form factor A 3 needs to be antisymmetric under the exchange of momenta p 1 ↔ p 2 .This leads to After setting A 3 = 0, the explicit form of the primary equations involving the form factor A 4 can be written as The solution is given by where ζ 2 is an arbitrary constant.Once again, due to the Bose symmetry, A 4 needs to be antisymmetric under the exchange of momenta p 1 ↔ p 2 .This leads to After setting A 4 = 0, we can write the remaining primary equations as These equations can also be reduced to a set of homogenous equations by repeatedly applying the operator The most general solution of such homogenous equations can be written in terms of the following 3K integrals where η i and θ i are arbitrary constants.The explicit form of such 3K integrals can be determined by following the procedure in [4,54,55].Before moving on, we need to examine the divergences in the 3K integrals.For a more detailed review of the topic, see Appendix A and [4,5,54].In general, it can be shown that a 3K integral If the above condition is satisfied, we need to regularize the integral.Therefore, we shift its parameters by small amounts proportional to a regulator ϵ according to the formula The arbitrary numbers u, v 1 , v 2 and v 3 specify the direction of the shift.In general the regulated integral exists, but exhibits singularities when ϵ is taken to zero.If a 3K integral in our solution diverges, we can expand the coefficient in front of such integral in the solution in powers of ϵ and then we can require that our entire solution is finite for ϵ → 0 by constraining the coefficients η (j) i and θ (j) i .Both of the 3K integrals appearing in the eq.( 99) in the solution for A 1 diverge like 1/ϵ.Therefore, we require Higher order terms do not contribute to the solution and therefore they can be neglected.In the case of A 2 , since some of the 3K integrals diverge like 1/ϵ 2 , we need to set The last step consists in analyzing all the conformal constraints on the numerical coefficients η (j) i and θ (j) i .In order to do that, we insert our solution back into the primary non-homogenous equations (97) and into the secondary equations.The explicit form of the secondary equations is given by2 We can solve all these equations by performing the limit p i → 0, as explained in the Appendix A.2.After some lengthy computations, using all the properties of the 3K integral listed in the Appendix A, we find that all the non-vanishing coefficients η (j) i and θ (j) i depend on the anomaly coefficient a 2 of eq. ( 14).In particular the final solution can be written in the compact form (106)

Perturbative realization
In this section we compute the ⟨T T J 5 ⟩ correlator perturbatevely at one-loop, working in the Breitenlohner-Maison scheme.For this analysis we shift to the Minkowski space where We consider the following action with a fermionic field in a gravitational and axial gauge field background where e µ a is the vielbein, e is its determinant and D µ is the covariant derivative defined as Σ ab are the generators of the Lorentz group in the case of a spin 1/2-field, while the spin connection is given by 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-express the action S 0 as follows Taking a first variation of the action with respect to the metric one can construct the energy momentum tensor as (113) 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.Their explicit expressions is reported in Appendix C.

Feynman diagrams
The ⟨T T J 5 ⟩ correlator around flat space is extracted by taking three functional derivatives of the effective action with respect to the metric and the gauge field, evaluated when the sources are turned off Having denoted with S 0 the conformal invariant classical action, recalling eq. ( 17), we can write where for the sake of simplicity we have used the notation g i = g µ i ν i (x i ) and A i = A µ i (x i ).The angle brackets denote the vacuum expectation value and each of the terms correspond to a Feynman diagram of specific topology.In particular, the first term has a triangle topology while the others are all bubble diagrams, except for the last one, which is a tadpole (see Fig. 1).The contribution of the triangle diagrams is given by (116) while the bubble diagrams are and After performing the integration, one can verify that V µ 1 ν 1 µ 2 ν 2 µ 3 2 vanishes.Lastly, the tadpole diagram is given by This last diagram vanishes since it contains the trace of two γ's and a γ 5 .The perturbative realization of the correlator will be written down as the sum of the amplitudes, formally given by the expression

Reconstruction of the correlator
The perturbative realization of the ⟨T T J 5 ⟩ satisfies the (anomalous) conservation and trace WIs.Therefore, the correlator can be decomposed as described in section 5.In particular, it is comprised of two terms The anomalous pole is given by which corresponds to eq. ( 61) with The transverse-traceless part ⟨t µ 1 ν 1 t µ 2 ν 2 j µ 3 5 ⟩ can be expressed in terms of four form factors as described in eq. ( 69).The perturbative calculation in four dimensions gives where C 0 in Minkowski space is the master integral with the Källen λ-function given by 8 Matching the perturbative solution In this section, we verify the matching between the perturbative form factors in Eq. ( 124) and the non-perturbative ones in Eq. ( 106).First of all, one can immediately see that A 3 and A 4 vanish in both calculations.On the other hand, in order to verify the matching between the first two form factors, we will need to rewrite the 3K integrals in the conformal solution in terms of the master integral C 0 .For this purpose, we recall the reduction relations presented in [54,55] Moreover, the integral I 1{0,0,0} is related to the massless scalar 1-loop 3-point momentum-space integral where Hence, it follows that the 3K integrals in our conformal solutions (106) are finite and can be reduced to By using the relations of the derivative acting on the master integral in Appendix D and setting the anomalous coefficient as in eq. ( 123), one can then verify the matching between the perturbative and non-perturbative form factors.
9 The anomaly pole of the gravitational anomaly and the sum rules As we have already mentioned in the previous sections, it is clear that our result does not depend on the specific expression of the current J 5 appearing in the correlator, since we have been using only the general symmetry properties of this 3-point function and its anomaly content in order to solve the conformal constraints.
Being the result unique and expressed in terms of a single constant, it shows that in a parity-odd CFT the gravitational anomaly vertex is generated by the exchange of an anomaly pole, with the entire correlator built around such massless pole and the value of its residue.Since this massless exchange was also present in the perturbative analysis of [34], we are now going to elaborate on those previous findings under the light of our current result.

Duality symmetry
The Maxwell equations in the absence of charges and currents satisfy the duality symmetry (E → B and B → −E).The symmetry can be viewed as a special case of a continuous symmetry where δβ is an infinitesimal SO(2) rotation and F µν = ϵ µνρσ F ρσ /2.Its finite form is indeed a symmetry of the equations of motion, but not of the Maxwell action.Notice that the action is invariant under an infinitesmal transformation modulo a total derivative.For β = π/2, the discrete case, then the action flips sign since (F 2 V → − F 2 V ).In general, the infinitesimal variation of the action takes the form Due to the equivalence (dual Bianchi identity) we can introduce the dual gauge field Ṽ µ which is related to the original V µ one by The current corresponding to the infinitesimal symmetry (135) can be expressed in the form whose conserved charge is gauge invariant after an integration by parts.Notice that the two terms on the equation above count the linking number of magnetic and electric lines respectively.In fluid mechanics, helicity is the volume integral of the scalar product of the velocity field with its curl given by and one recognizes in (140) the expression with B = ∇ × V and E = −∇ × Ṽ , that coincides with the optical helicity of the electromagnetic field [41].
As already mentioned, a perturbative analysis of ⟨T T J CS ⟩ has been presented long ago in [35].
The presence of anomaly poles in this correlator can indeed be extracted from [35], in agreement with our result.Indeed, for on-shell gravitons (g) and photons (γ), the authors obtain, with the inclusion of mass effects in the ⟨AV V ⟩, ⟨T T J f ⟩ and ⟨T T J CS ⟩ the following expressions for the matrix elements where q is the momentum of the chiral current.The anomaly poles are extracted by including a mass m in the propagators of the loop corrections, in the form of either a fermion mass for the ⟨AV V ⟩ and the ⟨T T J f ⟩, or working with a Proca spin-1 in the case of ⟨T T J CS ⟩, and then taking the limit for m → 0. A dispersive analysis gives for the corresponding spectral densities [35] ∆ AV V (q 2 , m) with v = 1 − 4m 2 /q 2 and d AV V = −1/2 α em , d T T J f = 1/(192π) and d T T J CS = 1/(96π) being the corresponding anomaly coefficients in the normalization of the currents of [35], with α em the electromagnetic coupling.Notice the different functional forms of ∆ T T J f (q 2 , m) and ∆ T T J CS (q 2 , m) away from the conformal limit, when the mass m is nonzero.One can easily check that in the massless limit the branch cut present in the previous spectral densities at q 2 = 4m 2 turns into a pole lim m→0 ∆(q 2 , m) ∝ δ(q 2 ) (147) in all the three cases.Beside, one can easily show that the same spectral densities satisfy three sum rules indicating that for any deformation m from the conformal limit, the integral under ∆(s, m) is mass independent.Therefore, the numerical value of the area equals the value of the anomaly coefficient in each case.One can verify, from equation (106), by taking the on-shell photon/graviton limit, that the transverse sector of ⟨T T J 5 ⟩, corresponding to the form factors A 1 and A 2 , vanishes, since these two form factors are zero, limiting each of these matrix elements to only single form factors, as indicated in (144) and (145).Then it is clear that, in general, the structure of the anomaly action responsible for the generation of the gravitational chiral anomaly can be expressed in the form where the ellipses stand for the transverse sector, and A λ is a spin-1 external source.For on-shell gravitons, as remarked above, this action summarizes the effect of the entire chiral gravitational anomaly vertex, being exactly given by the exchange of a single anomaly pole.

Summary of the results
Before coming to our comments and conclusions, for the reader's convenience, we briefly summarize our findings.
We have shown that in a general CFT the ⟨T T J 5 ⟩ correlator can be written as a sum of two terms the first term being the transverse component and the second, the longitudinal one, expressed in terms of a single anomaly form factor and tensor structure.This is characterized by an interpolating anomaly pole.The anomaly part is given by the expression (153) while the transverse-traceless part is with A 1 and A 2 given by eq.( 106).The entire correlator is therefore determined only by the anomalous coefficient a 2 in (153).
We have also computed the correlator perturbatively at one-loop in free field theory and verified the agreement of the expression with the non-perturbative results obtained by imposing the conformal symmetry.The explicit expressions of the form factors A 1 and A 2 have been given in (124).The solutions of the conformal constraints, expressed in terms of 3K integrals I 5{2,1,1} and I 4{2,1,0} , can be related to the ordinary one-loop master integrals C 0 and B 0 by (131).They can be reconstructed using recursively the relations included in Appendix D.
11 Comments: non-renormalization of the ⟨AV V ⟩ and ⟨T T J 5 ⟩ and the soft photon/graviton limits Before coming to our conclusions, we pause for few comments on the results of our paper, in relation to our previous study of the ⟨AV V ⟩ chiral anomaly vertex, in a general CP-violating CFT [32].
In the case of the ⟨AV V ⟩ vertex, the Adler-Bardeen theorem shows that the longitudinal part of the interaction is not affected by renormalization and therefore can be computed exactly just from the one-loop triangle diagram, being protected from perturbative corrections at higher orders.This is not true for the transverse part of the same diagram, that satisfies an ordinary WI.However, in [56] it was pointed out that, in the kinematic limit where the momentum of one of the vector currents is vanishingly small, another non-renormalization theorem is valid.Indeed, in that limit just two independent form factors are needed to fully describe the ⟨AV V ⟩ correlator.One of these form factors is related to the axial anomaly and, therefore, it is not renormalized.The other form factor belongs to the transverse sector.
In [56] it was shown that, due to helicity conservation in massless QCD, the two form factors are in fact proportional to each other, and so the non-renormalization of one of them implies that of the other.If the anomalous behaviour is identified with the exchange of an anomaly pole, that result relates the anomaly pole to the transverse part of the diagram, when one of the photons becomes soft.
Perturbative analysis of the diagram -in the most general kinematics -showed that at two-loops the entire diagram is indeed non renormalized [57], a feature that disappears at higher perturbative orders.Indeed, the authors of [58] found non-vanishing corrections to the correlator at O(α 2 S ).The non-renormalization to all orders of a specific combination of the transverse form factors of the ⟨AV V ⟩ was shown to hold in [59], in the chiral limit of QCD, since it equals the longitudinal form factor.The latter is, obviously, non-renormalized since the anomaly pole and its residue are protected.Other combinations of purely transverse form factors were also shown to be nonrenormalzed.In our anaysis [32] we have shown -just using the conformal constraints -that such results indeed follow from conformal symmetry, once these are solved either in the most general kinematics or in the specific one required by Vainshtein's conjecture [56].Therefore, the breaking of the nonrenormalizaton theorem for the entire vertex in QCD must originate from terms breaking conformal invariance and must be proportional to the QCD β function.
In this paper we have verified that a similar connection between the longitudinal and the transverse part is present in the case of the ⟨T T J 5 ⟩ correlator in the conformal limit, being both sectors proportional to the a 2 anomalous coefficient.With these new indications, that follow quite closely the ⟨AV V ⟩ case previously discussed by us, it would be interesting to test, at the perturbative level, if in the soft graviton limit a similar result holds for this correlator at all orders in perturbative QCD.We do expect that the higher order corrections will be proportional to the QCD β function, therefore breaking the conformal symmetry.

Conclusions
We have presented an analysis of the gravitational anomaly vertex from the perspective of CFT in momentum space.We have shown how the vertex can be completely defined by the inclusion of a single anomaly pole together with the CWIs.This explicit analysis shows that reconstruction method formulated in the parity-even sector in the case of conformal anomaly correlators can be extended quite naturally to the parity-odd sector.This provides a different and complementary perspective on the origin of anomalies and their related effective actions, which may account for such phenomena.This extension highlights the intrinsic connection between these seemingly distinct sectors and suggests a unified framework for comprehending the origin of anomalies.It underscores the notion that anomalies, whether chiral, conformal, or supersymmetric, share a common underlying structure characterized by the presence of a single (anomaly) form factor, together with a specific tensor structure responsible for generating the anomaly.The approach does not rely on the explicit structure of the parity-odd current appearing in the correlator but, rather, on its symmetry properties.We have also shown that, similarly to previous dispersive analysis of the anomalous form factors for the ⟨T JJ⟩ and ⟨AV V ⟩ diagrams, the spectral density of the anomalous form factor of the ⟨T T J 5 ⟩ satisfies a sum rule.The numerical value of the sum rule is fixed by the anomaly.
The triple-K integral depends on four parameters: the power α of the integration variable x, and the three Bessel function indices β j .The arguments of the 3K integral are magnitudes of momenta p j with j = 1, 2, 3.One can notice the integral is invariant under the exchange (p j , β j ) ↔ (p i , β i ).We will also use the reduced version of the 3K integral defined as where we introduced the condensed notation {k j } = {k 1 , k 2 , k 3 }.The 3K integral satisfies an equation analogous to the dilatation equation with scaling degree where From this analysis, it is simple to relate the form factors to the 3K integrals.Indeed, the dilatation WI of each from factor tells us that this needs to be written as a combination of integrals of the following type where N is the number of momenta that the form factor multiplies in the decomposition.Let us now list some useful properties of 3K integrals

A.2 Zero momentum limit
When solving the secondary CWIs, it may be useful to perform a zero momentum limit.In this subsection, we review the behaviour of the 3K integrals in the limit p 3 → 0. In this limit, the momentum conservation gives Assuming that α > β t − 1 and β 3 > 0, we can write lim where We can derive similar formulas for the case p 1 → 0 or p 2 → 0 by considering the fact that 3K integrals are invariant under the exchange (p j , β j ) ↔ (p i , β i ).

A.3 Divergences and regularization
The 3K integral defined in (155) converges when If α does not satisfy this inequality, the integrals must be defined by an analytic continuation.The quantity is the expected degree of divergence.However, when for some non-negative integer k and any choice of the ± sign, the analytic continuation of the 3K integral generally has poles in the regularization parameter.Therefore, if the above condition is satisfied, we need to regularize the integrals.This can be done by shifting the parameters of the 3K integrals as or equivalently by considering In general, the regularisation parameters u and v i are arbitrary.However, in certain cases, there may be some constraints on them.For simplicity, in this paper we consider the same v i = v for every i.

A.4 3K integrals and Feynman integrals
3K integrals are related to Feynman integrals in momentum space.The exact relations were first derived in [4,54].Here we briefly show the results.Such expressions have been recently used in order to show the connection between the conformal analysis and the perturbative one for the ⟨AV V ⟩ correlator [32].Let K d{δ 1 δ 2 δ 3 } denote a massless scalar 1-loop 3-point momentum space integral Any such integral can be expressed in terms of 3K integrals and vice versa.For scalar integrals the relation reads where β t = β 1 + β 2 + β 3 .All tensorial massless 1-loop 3-point momentum-space integrals can also be expressed in terms of a number of 3K integrals when their tensorial structure is resolved by standard methods (for the exact expressions in this case see Appendix A.3 of [4]).

Figure 1 :
Figure 1: Feynman Diagrams of the three different topologies appearing in the perturbative computation.