Parity-odd 3-point functions from CFT in momentum space and the chiral anomaly

We illustrate how the Conformal Ward Identities (CWI) in momentum space for parity-odd correlators determine the structure of a chiral anomaly interaction, taking the example of the VVA (vector/vector/axial-vector) and AAA correlators in momentum space. Only the conservation and the anomalous WIs, together with the Bose symmetry, are imposed from the outset for the determination of the correlators. We use a longitudinal/transverse decomposition of tensor structures and form factors. The longitudinal (L) component is fixed by the anomaly content and the anomaly pole, while in the transverse (T) sector we define a new parameterization. We relate the latter both to the Rosenberg original representation of the VVA and to the longitudinal/transverse (L/T) one, first introduced in the analysis of g-2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g-2$$\end{document} of the muon in the investigation of the diagram in the chiral limit of QCD. The correlators are completely identified by the conformal constraints whose solutions are fixed only by the anomaly coefficient, the residue of the anomaly pole. In both cases, our CFT result matches the one-loop perturbative expression, as expected. The CWIs for correlators of mixed chirality JLJJR\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$J_L J J_R$$\end{document} generate solutions in agreement with the all-orders nonrenormalization theorems of perturbative QCD and in the chiral limit.


Introduction
Parity even correlators in d = 4 play a central role in CFT and have been investigated in coordinate space [1,2] for quite some time.Parity odd ones, instead, have attracted less attention, except for the JJJ 5 and J 5 J 5 J 5 (or V V A and AAA, where A and V refer to the axial-vector and to the vector current respectively) due to the role of the chiral anomaly.The conformal properties of the VVA diagram, in the coordinate space, have been studied since the 70's (see for instance [3] and [4]).In more recent years, its non-renormalization property, which affects only its anomaly (longitudinal) part, as shown by Adler and Bardeen [5], has been redrawn in a more general context, given its relevance in the analysis of g − 2 of the muon [6].The numerical value of a µ = (g − 2) µ = F M (0), which measures the muon anomaly, is given by the Pauli form factor at zero momentum transfer, and involves a soft photon limit on one of the two vector currents of the V V A diagram, which interpolates with F M via electroweak corrections.In such special kinematic configuration, the correlator is described by the longitudinal and transverse components w L and w T of the diagram.If we adopt such a L/T separation of the vertex, w L does not receive radiative corrections, as shown in [5].Therefore, any constraining relation involving both w L and w T is bound to identify combinations of form factors in the T sector, which are protected against radiative corrections.The conditions under which such nonrenormalization constraints hold, may involve a special kinematics.For this reason, the analysis of the correlator requires moving into momentum space.It has been argued that in the chiral limit of QCD and with a soft V line, the relation holds to all orders in the strong coupling constant α s [6].This constraint is indeed reproduced by the bare fermion loop, which is conformal, and is a byproduct of our analysis of the CWIs in momentum space as well.Notice that, given the complete overlap of the CFT result with the perturbative one at the lowest order, identities such as the Crewther-Broadhurst-Kataev (CBK) relation [7,8,9], are exactly satisfied in our case as well (see [10]).The CBK identity connects the Adler function for electron-positron annihilation with the Bjorken function of deep-inelastic sum rules.Away from the conformal limit, these relations acquire corrections proportional to the QCD β function at higher orders.A perturbative analysis of the radiative corrections of the V V A correlator in QCD at two-loop level (O(α s ), showed that the entire vertex is also non renormalized [11], in agreement with (1).However, at higher orders in α s (O(α 2 s )), the nonrenormalization properties of the entire diagram are violated and conformality is lost.In particular (1) is only limited to the soft photon limit discussed in [6], as shown in explicit computations [12].In our analysis, the constraint in (1) can be viewed also as a result of conformal symmetry, without resorting to a perturbative picture, since in the unique solution of the CWIs from momentum space, such constraint is verified.While the result is expected, the procedure is novel.One interesting aspect of the identification of the V V A or AAA diagrams from the CWIs in momentum space, is the centrality of the anomaly pole in the longitudinal sector, that allows to identify every part of the correlator.The link between such massless state and the nonlocal anomaly effective action has been stressed in the past in several works, [13] [14] [15].More recently, these analysis have been extended in the investigation of chiral density waves, with local actions [16] [17].These actions play a key role in very different contexts where either chiral or conformal anomalies are involved, such as in topological materials [18] [19].

Conformal analysis in momentum space
In coordinate space the identification of the structure of conformal correlators is quite direct, but does not provide much information on the dynamical properties of those, among them, which are affected by an anomaly.In momentum space the vertices of such interactions have been investigated, in perturbation theory, in terms of different (minimal and non minimal) sets of form factors, whose expressions can be calculated at one-loop level by standard perturbative methods.At d = 4 the Schouten identities play an important role in the choice of the most useful representations of such diagrams, and relate the form factors of the different parameterizations.For instance, the w L /w T decomposition, that plays a fundamental role in the investigations mentioned above, can be related to other representations of the corresponding form factors. Being a free fermion Lagrangian at d = 4 conformal at tree level, the chiral anomaly fermion loop provides a simple free field theory realization of the VVA vertex, which is also conformal.As we have mentioned, the conformal symmetry is eventually broken by radiative corrections only at order α 2 s , in the strong coupling constant.Anomalies arise from the region of the correlator where all the external coordinate points (x, y, z) coalesce, and for this reason, by a Fourier transform, their analysis in momentum space has the advantage of including this region rather automatically.While the conformal contributions to such correlators at d = 4 are described, as mentioned, by the simple massless fermion loop, the identification of the kinematical structure of the vertex in momentum space -without any reference to a Feynman diagram realization and using only the Ward identities (WIs) of the case -has never been discussed before.This is the goal of our work.We are going to close this gap and show how the entire correlator is fixed by the anomaly also if we resort to a momentum space analysis.We will be using a variant of the separation of the CWIs into primary and secondary equations [20], in which the anomaly constraint on the longitudinal form factor is imposed from the beginning.This approach differs from the cases discussed so far for 3-point and 4-point functions affected by the conformal anomaly, since in that case the anomaly emerges from the renormalization procedure, realized by the inclusion of a Weyl squared counterterm after renormalization (see [21,22] for a Lagrangian formulation).Renormalized parity even correlators, type-A and type-B Weyl anomalies have been discussed in [23,24].It is well known that for a VVA vertex, as for any anomaly interaction which is purely topological, the process of renormalization can be entirely bypassed by imposing the anomalous WIs on the external currents.Our approach extends to chirally odd three-point functions the methods used in the analysis of the parity even ones in momentum space [20,25], in the presence of a chiral anomaly, and can be formulated also for higher point functions.Recently novel approaches have been adopted for the construction of parity odd correlation functions.In particular in [26] it is shown that parity odd CFT 3-point functions can be obtained by doing epsilon transformation starting from parity even CFT correlation functions.Moreover, in [27] the authors use both the momentum space CWIs as well as spin-raising and weight-shifting operators to fix the form of parity odd correlators.However both [26] and [27] do not consider anomalous correlators.A recent analysis of 3-point functions for parity even correlators with non-conserved currents of arbitrary spin has been discussed in [28].Moreover, another recent analysis on the conformal bootstrap equations in momentum space at finite volume has been discussed in [29].

Massless intermediate states in the anomaly: the pivot
The construction of the entire correlator proceeds from the anomaly pole, that plays a key role in any anomaly diagram.This takes the role of a pivot in the procedure, and it allows to solve for the longitudinal anomalous WI quite straightforwardly.
The tensorial expansion of a chiral vertex is not unique, due to the presence of Schouten relations among its tensor components.For instance, an anomaly pole in the virtuality of the axial-vector current -denoted as 1/p 2 3 in our notations -can be inserted or removed from a given tensorial decomposition, just by the use of these relations.These identities connect two of the most common representation of this vertex, the first of them introduced long ago by Rosenberg [30], expressed in terms of 6 tensor structures and form factors, and the second one [31], more recent, introduced in the context of the analysis of the g − 2 anomalous magnetic moment of the muon.The latter, which is the most valuable from the physical point of view, allows to attribute the anomaly to the exchange of a pole in the longitudinal channel [32] [13, 33].In this second parameterization of the vertex, the decomposition identifies longitudinal and transverse components where W T is the transverse part, while the longitudinal tensor structure is given by w L is the anomaly form factor, that in the massless (chiral or conformal) has a 1/p 2 3 pole.In the case of gauge anomalies, the cancelation of the anomaly poles is entirely connected with the particle content of the theory and defines the condition for the elimination of such massless interactions.The total residue at the pole then identifies the total anomaly of a certain fermion multiplet.A similar behaviour holds for conformal correlators with stress energy tensors, where the residue at the pole coincides with the β-function of the Lagrangian field theory, and is determined by the number of massless degrees of freedom included in the corresponding anomaly vertex, at the scale at which the perturbative prescription holds [15].As illustrated in Fig. 1, the pole emerges from the region of integration in momentum space in the VVA loop when an effective interaction, mediated by the fermion/antifermion pair is generated.The two collinear (on-shell) particles describe an effective pseudoscalar interaction interpolating between the incoming axial vector and the two outgoing vector currents.Away from the conformal limit, when the VVA diagram is recomputed with the inclusion of a fermion of mass m, one discovers the presence of a sum rule satisfied by the spectral density of the form factor w L of the longitudinal channel.Such form factor (w L ) is characterised by a spectral density ρ(s), whose integration in the region 4m 2 < s < ∞ in the dispersive variable s, related to the virtuality of the axial-vector current, is mass-independent and given by where a is the anomaly for a single Dirac fermion.For on-shell vector lines the perturbative vertex reduces only to the longitudinal component W L .
The separation into transverse and longitudinal contributions appears to be ambiguous, in the sense that one could always add and subtract transverse contributions to the pole part.However, one can show that such 1/p 2 3 behaviour does not depend on the parameterization of the fermion loop.This important point is illustrated in the case of the longitudinal transverse (L/T) representation, once this representation is mapped to the Rosenberg representation, as we are going to comment below.A perturbative reparameterization of the loop momentum in the anomaly diagram is equivalent to the inclusion of Chern-Simons terms in the representation of two of the six form factors (B 1 and B 2 ) that we are going to define in the next sections, but it is independent of them in the second parameterization (w L ∼ B 1 − B 2 ).Our analysis shows the centrality of the pole in determining the general solution of the CWIs, that is only fixed by the anomaly.Some of these points are reviewed in the appendix for completeness.The procedure follows a decomposition similar to the one used in the case of correlators of stress energy tensors.It is based on the parameterization of such correlators in terms of a transverse traceless and a longitudinal sector.In the conformal case the trace anomaly sector of the correlator, identified by pole structures similar to the one that we are going to discuss here, needs to be modified by the inclusion of traceless Weyl invariant terms [34,35].These are necessary to decompose the correlator into two traceless and trace parts which both satisfy energy momentum conservation.In the case of the chiral VVA the procedure is far simpler.In the case of the AAA diagram, as we are going to see, we are going to encounter contributions which are proportional to Chern-Simons (CS) forms.These are terms linear in two of the three momenta and allow to move the anomaly form one vertex to the other.For instance, in a perturbative evaluation of the V V A and AAA diagrams, in the chiral limit, the anomaly can be moved around the vertices by the inclusion of such CS terms The term ε αµ 1 µ 2 µ 3 (p 1 − p 2 ) α corresponds to the Chern-Simons contribution.
In our analysis we proceed from the V V A case, then turning to the AAA and show in both cases how primary and secondary CWI can be solved quite efficiently in momentum space.As shown in Fig. 2, the structure of the AAA is dictated by symmetry.One can move around the anomaly content, from the AAA with CS terms in order to obtain the VVA [36].Our solutions are then compared with the two most common representations of such diagrams, the Rosenberg and the L/T one.The latter is particularly useful for the role that provides for the longitudinal component of the correlator.The same representation has been used in the past for the derivation of new nonrenormalization theorems in perturbative QCD, in the chiral limit.This result has been derived in [31] and is confirmed by the formal analysis of the CWIs, once we turn to discuss the relation between an AV V and a V V A diagram, whose difference, in our analysis, is found to be expressed in terms of a CS form.For convenience, we have included an appendix where some of the more technical aspects of our derivation are collected.The analysis of the conformal constraints for ⟨JJJ 5 ⟩, as already mentioned, is performed by applying the L/T decomposition to the correlator.We focus our analysis on the d = 4 case, where the conformal dimensions of the conserved currents J µ are ∆ = 3 and the tensorial structures of the correlator will involve the antisymmetric tensor in four dimensions ϵ µναβ .The procedure to obtain the general structure of the correlator starts from the conservation Ward identities of the expectation value of the non anomalous J µ and anomalous J µ 5 currents.The vector currents are coupled to the vector source V µ and the axial-vector current to the source A µ .Applying multiple functional derivatives to (6) with respect to the source V µ , after a Fourier transform, we find the conservation Ward identities related to the entire correlator which are given by where ε p 1 p 2 µ 1 µ 2 ≡ ε αβµ 1 µ 2 p 1α p 2β and the momenta are all incoming.From this relations we construct the general form of the correlator, splitting the operators into a transverse and a longitudinal part as J µ (p) = j µ (p) + j µ loc (p), and Due to (7), the correlator is purely transverse in the vector currents.We then have the following decomposition where the first term is completely transverse with respect to the momenta p iµ i , i = 1, 2, 3 and the second term is the longitudinal part that is proper of the anomaly contribution.Using the anomaly constraint on j 5loc we obtain On the other hand, the transverse part can be formally written as where we have made a choice on which independent momenta to consider for each index, and in particular The correlator has to be symmetric under the exchange of the two vector currents and this fact is reflected in the symmetry constraints reducing by two the number of independent form factors. Furthermore, in d = 4 a class of tensor identities has to be considered, for instance the Schouten identity From this type of tensor identities we find that reducing the number of independent form factors just to two.We conclude that the general structure of the transverse part is given by where

Dilatation Ward identities
We now start to analyse the conformal constraints on the form factors.In this and the next sections, we closely follows the methodology adopted in [20].The invariance of the correlator under dilatation is reflected in the equation Considering the decomposition (10) in the previous equation, then the transverse part of the correlator has to satisfy By using the chain rule in term of the invariants for the form factors in order to obtain with N j the number of momenta that the form factors multiply in the decomposition ( 18) and then we have N 1 = 3 and N 2 = 1.

Special Conformal Ward identities
The invariance of the correlator with respect to the special conformal transformations is encoded in the special conformal Ward identities The special conformal operator K κ acts as an endomorphism on the transverse sector of the entire correlator.Therefore we can perform a transverse projection (3 − π projection) on all the indices in order to identify a set of partial differential equations corresponding to the primary constraints [20] splitting the correlator into its transverse and longitudinal parts.Using the Schouten identities listed in Appendix A, we decompose the action of the special conformal operator on the transverse sector using the transverse projectors.One can show that the action of the operator is endomorphic in the transverse sector.Therefore we have where the tensor X κµ 1 µ 2 µ 3 can be constructed using a set of possible tensor structures, of the form whose complete list is given in Appendix A. These tensor structures are not all independent, and are simplified using some Schouten identites in order to find the minimal number of tensor structures in which X can be expanded.The first two identities are that can be contracted with p 1α and p 2α to generate, after the projection, tensor identities of the form More technical details and a full list of such tensor relations is given in the Appendix.Combining all the expressions we obtain the projected equation where the C ij are scalar differential equations involving the form factors.In particular, C 1j and C 2j are of the second order, while all the others are first order differential equations.The action of K κ on the longitudinal part is then given as where is related to the chiral anomaly.Due to the independence of the tensor structures listed above, the special conformal equations are written as the explicit form of the primary equations ( 34) is where we have defined These equations can also be reduced to a set of homogenous equations by repeatedly applying the operator K ij and we have

Solutions of the CWIs
The most general solution of the conformal Ward identities of the V V A can be written in terms of integrals involving a product of three Bessel functions, namely 3K integrals [20].For a detailed review on the properties of such integrals, see also [37].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 triple-K 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 [20] deg where From this analysis, it is simple to relate the form factors to the triple-K integrals.Indeed, the dilatation Ward identities tell us that the form factor A i needs to be written as a combination of integrals of the following type where N i is the number of momenta that the form factor multiplies in the decomposition (18).The special conformal Ward identities fix the remaining indices k 1 , k 2 and k 3 .Indeed, recalling the following property of the 3K integrals we can write the most general solution of the homogeneous equations (41) as Applying this general solutions to the non-homogenous equations (39), we find the constraint and the solution reduces to We now consider the first order differential equations ( 35), ( 36), ( 37) and (38) in their explicit form, ignoring the trivial ones.We start with and using the property of the 3K integral we have obtaining the constraint Taking into account this constraint, we consider the other secondary Ward identity as and this equation with the constraints obtained before on the coefficients α i is trivially satisfied.
The last equation we have to consider is the one related to the anomaly that takes the form This equation can be easily solved by taking the limit p 3µ → 0, p 1µ = −p 2µ = p µ .Assuming that α > β t − 1 and β 3 > 0, we can write [20] lim and then lim where (63) With this limit the equation can be solved and we have, with d = 4 and ∆ 3 = 3, the constraint In summary, once the conformal constraints are solved, we find the solution of the transverse part in terms of one coefficient, proportional to the anomaly and in particular we have or in terms of the simplified version of the 3K integrals as Explicitly we have Furthermore, this expression can be reduced by noticing that finally giving for (67) 3 Reducing the 3K integral in the solution Using the reduction relations presented in [37,38] we have that the solution is finite and can be reduced to the standard perturbation results as where I 1{0,0,0} is the master integral related to the three-point function of the operator φ 2 in the theory of free massless scalar φ in d = 4. Indeed, the relation between the master integrals and the massless scalar 1-loop 3-point momentum space integral is where By using the relations of the derivative acting on the master integral presented in [21], and analytically continuing C 0 to d dimensions we find that where λ is the Källen λ-function This expression is finite in the expansion around d = 4 and the result simplifies to the form with the form factor given by 3 ). (78)

Perturbative realization of the correlator
The same approach that we have investigated in the previous section, can be redone in the context of a free field theory, with a single chiral fermion.The steps are the same as above, with the correlator expanded in the same basis of form factors identified above using ( 10) and ( 12), using a free Dirac fermion and the only form factor A 2 is given by On can directly check the complete match between the perturbative and the CFT result, once the anomaly coefficient is chosen of the form 4 The conformal ⟨AAA⟩ correlator In this section we illustrate how conformal invariance completely determines the structure of the ⟨J 5 J 5 J 5 ⟩ correlator.Differently from the ⟨JJJ 5 ⟩ correlator, a 3K integral regularization is needed in this case.Therefore, we will work in d = 4 + ϵ and then perform the limit ϵ → 0.

Longitudinal/Transverse decomposition
First of all, we consider the anomalous Ward identity where F A µν is the gauge field coupled to the axial current.We impose such identity symmetrically on the all the three external axial-vector currents of the AAA correlator, leading to the following equations Notice that the equations above are symmetric in the three momenta, but for technical reasons we prefer to express them only in terms of p 1 and p 2 .Note that, if we contract the correlator with multiple momenta at the same time, the result is zero.We can then decompose the correlator into the following transverse and longitudinal parts The longitudinal parts are completely fixed by the anomalous Ward identity above in the form On the other hand, the transverse part can be expressed as Differently from the ⟨JJJ 5 ⟩ correlator, there is an additional Bose symmetry condition we have to consider: {p 1 , µ 1 } ↔ {p 3 , µ 3 }.Therefore, in this case, there is only one independent form factor. Indeed we have For now on we will ignore such relations and we will treat Ã and A as independent quantities.We can still check later that our final result is invariant under the exchange of the currents.

Dilatation and Special Conformal Ward identities
The dilatation Ward identities of the transverse part are not affected by the longitudinal terms.Therefore the constraints are the same of the ⟨JJJ 5 ⟩ correlator The invariance of the correlator with respect to the special conformal transformations is encoded in the following relation .
(89) We procede in a manner similar to the ⟨JJJ 5 ⟩ correlator, using the same Schouten identities.We can then decompose the contribution of the transverse part into the following form factors where the explicit expression for the form factor is the same of the ⟨JJJ 5 ⟩.However, both the primary and secondary equations will contain anomalous terms from the longitudinal parts of the correlators.Indeed, the primary Ward identities are given by while the secondary equations can be written as (91) The explicit form of the primary special conformal Ward identity is (92) These equations can also be reduced to a set of homogenous equations by repeatedly applying the operator As we can see, the presence of anomalous terms in the primary equations does not affect the structure of the homogenous equations which are exactly the same of the ⟨JJJ 5 ⟩ correlator.

Solutions of the CWIs
The solutions of the primary homogeneous equations can be written in terms of the following 3K integrals, extending the previous approach of the V V A Inserting our solutions back to the non-homogeneous equations, in the limit p 3 → 0 we find We then focus on the first two secondary equations in (91).The explicit form of the first equation is which leads to the condition The explicit form of the second equation in ( 91) is which leads to the constraint The other secondary equations don't impose any other constraints.We insert such conditions into our solution and use the following property of the 3K integral in order to arrive to the following expression in d = 4 + ϵ Note that we are keeping the second term of A because the 3K integral has a pole in ϵ.In the end we have

Connection with the ⟨V V A⟩ correlator
Recalling that a ′ = a/3, our results are in accordance with the formula In order to prove such formula we use the relation At this stage, exchanging the second current with the third, we have while, exchanging the first current with the third one, we obtain Summing the last three equations and using the following identity of the 3K integrals we arrive at Eq. ( 105).

Comparison with other parameterizations
Having established the agreement between the perturbative (lowest order) and the non-perturbative computation of the correlator, we try to relate the result of the expansion introduced in ( 12) with the two most popular parameterizations of the same vertex.
As we have already mentioned in the introduction, the original parameterization of the V V A was presented in [30].Lorentz symmetry and parity fix the correlation function in the form with B 1 and B 2 divergent by power counting.If we use a diagrammatic evaluation of the correlator, the four invariant amplitudes B i for i ≥ 3 are given by explicit parametric integrals [30] where the general massive I st integral is defined by Both B 1 and B 2 can be rendered finite by imposing the Ward identities on the two vector lines, giving Table 1: Tensor structures of odd parity in the expansion of the V V A with conserved vector currents.
which allow to re-express the formally divergent amplitudes in terms of the convergent ones.The Bose symmetry on the two vector vertices is fulfilled thanks to the relations Using the conservation WIs for the vector currents, we obtain the convergent expansion [39] where in the last step we have introduced four tensor structures that are mapped into one another under the Bose symmetry of the two vector lines.One can identifies six of them, as indicated in Table 1, but two of them are related by the Schouten relations to the other four, η 3 , . . .η 6 .Indeed one has The remaining tensor structures are inter-related by the Bose symmetry The correct counting of the independent form factors/tensor structures can be done only after we split each of them into their symmetric and antisymmetric components where we omitted all the tensorial indices which are in the order µ 1 µ 2 µ 3 .We can then re-express the correlator as

L/T decomposition
An alternative parameterization of the V V A correlator, which allows to set a direct comparison with the one that we have introduced in the previous sections is given by [31] where the longitudinal component is specified in eq. ( 3), while the transverse component is given by This decomposition automatically account for all the symmetries of the correlator with the transverse tensors given by The map between the Rosenberg representation and the current one is given by the relations and viceversa or, after the imposition of the Ward identities in Eqs.(114,115), where B i ≡ B i (p 1 , p 2 ).As already mentioned, (130) is a special relation, since it shows that the pole is not affected by Chern-Simons forms, telling us of the physical character of this part of the interaction.
Also in this case, the counting of the form factor is four, one for the longitudinal pole part and 3 for the transverse part.Notice that all of them are either symmetric or antisymmetric by construction.
To relate this decomposition to our, we apply the transverse projectors and obtain and by using the Schouten identities we obtain We then identify the form factors of our decomposition and the current L/T one in the form Notice that A 1 is antisymmetric in the exchange of the two vector lines and counts for one independent form factor, while A 2 contains both symmetric and antisymmetric components and counts as two.Combined with w L , we again find that our form factors are four in the general case, before enforcing the conformal WIs on the parameterization.One can check from the solution of the special CWIs that this number is reduced by one in both representations, since in this case for the L/T one.In our case the form factor A 1 vanishes and we are left with three form factors in both cases.Proceeding in a similar manner, we can also map the Rosenberg parametrization into the one we worked in.The results are 6 Nonrenormalization theorems In this section we establish a connection between the conformal solutions of correlators of mixed chirality and the perturbative results previously derived for them in the chiral limit of QCD.They take the form of non-renormalization theorems, originally presented in [31].They are a direct consequence of the fact that correlators of currents of different chiralities vanish in such limit.Also in this case one identifies constraints between w L , the pole part, and combinations of transverse form factors, which are not affected by radiative corrections.
From the perturbative picture, these theorems can be understood quite easily at diagrammatic level.The reason lays in the absence of explicit mass insertions in the perturbative expansion of a ⟨J L JJ R ⟩ correlator, if computed in the chiral limit.In this case, chirality flips on the fermion lines of the the contour of the diagram in Fig. (1) are prohibited, and the vector-like nature of the QCD interactions with the fermions does the rest, guaranteeing the vanishing of the correlator.As just mentioned, the theorems in [31] are derived from the ⟨J L JJ R ⟩ vertex, where J L and J R are left and right chiral currents while J is vector-like.The building block of ⟨J L JJ R ⟩ is the JJJ 5 (or V V A) correlator.All the other diagrams, in the chiral limit, such as the AV V or V AV or AAA, are trivially related to the V V A due to the anticommuting property of γ 5 and the symmetry constrain as illustrated in Fig. 2. In perturbation theory, the anomaly (a ′ ) content of the AAA, for instance, can be determined on the basis of symmetry, assuming an equal sharing (a/3) of the anomaly for each external axial-vector line.The constraint can be used as a starting point for moving the anomaly around the vertices, by the inclusion of appropriate Chern-Simons terms.In the Rosenberg representation [30] they amount to shifts of the form factors B 1 and B 2 (see also the appendix).
In [31], the authors analyzed the ⟨J L JJ R ⟩ correlator Using the charge conjugation invariance, they set the parity-even contribution ⟨V V V ⟩ and ⟨AV A⟩ to zero.Alternatively, we can assume the correlator is conformally invariant and arrive to the same conclusion [20].We start from this point.
Conformal invariance requires the abelian even-parity correlator ⟨JJJ⟩ to be zero in any dimension.
In order to show that, first we impose the following conservation Ward identities Therefore the correlator is purely transverse and can be expressed as The A 1 form factor is completely antisymmetric for any permutation of the momenta while the form factor A 2 is antisymmetric under (p 1 ↔ p 2 ).We now consider the conformal constraints on the form factors.The dilatation Ward identities are The primary special conformal Ward identities are The solution to such equations can be written in terms of the following 3K integrals The secondary special conformal Ward identities are where we defined the following operators Inserting our solution (151) into such equations, we arrive to the conditions α 1 = α 2 = 0. Since ⟨AV A⟩ is not anomalous, one can prove in the same way that this correlator vanishes too.Therefore, we are left with The authors of [31] assumed that the ⟨J L JJ R ⟩ correlator is simply a Chern-Simons term, in order to prove the nonrenormalization theorems.Using our conformal solution, we can directly write the expression for the ⟨J L JJ R ⟩ correlator and prove their statement.Indeed for the longitudinal part we can write while for the transverse part, after contracting the projectors in our solution, we have (156) where in the last row we used the explicit expression of the 3K integral.Adding the contributions together, we arrive to which tells us that the ⟨J L JJ R ⟩ correlator is simply given by a Chern-Simons term, as expected.Note that, proceeding in a similar manner, one can prove eq. ( 5) too.
Of course, one can also check that our conformal solution directly satisfies the following nonrenormalization theorems, originally derived in the chiral limit of perturbative QCD 0 = w

Conclusions
In this work we have illustrated how the CWIs in momentum space can be used to determine the structure of chiral anomaly diagrams in an autonomous way respect to coordinate space.This shows that anomalies in CFT can be treated consistently in this specific framework, that allows to establish a link between such correlators and the ordinary perturbative amplitudes.Parityodd correlators are important in many physical context, and in this case we have shown how the conformal properties of previous perturbative analysis, performed in free field theory in the chiral limit, are the result of conformal symmetry and of its constraints.Our derivation does not rely on any Lagrangian realization.The analysis of parity odd correlators, especially for 4-point functions, is still under investigation, given its complexity, and the inclusion of the anomaly content is for certainly an interesting aspect that deserves a closer attention.It may be possible in the future to consider mixed conformal/chiral anomalies in the same framework.We hope to come back to the investigation of these points in future work.
that can be contracted with p 1α and p 2α and taking the projectors in front we obtain the four tensor identities Then, considering the identity and contracting with δ κ α , p 1α and p 2α we find Furthermore, we need to consider the identity that once it is contracted with p 1α and p 2α we have and it is worth mentioning that the possible contraction with δ µ 3 α give again an identity that is not independent taking in consideration the previous tensor identities found.

B Discontinuities in 3-point functions
The presence of a sum rule related to the anomaly is connected with the behaviour of the spectral density for the triangle diagram and its scalar 3-point function C 0 with the spectral density computed from its discontinuity with the usual iϵ prescription (ϵ > 0) We can use the unitarity cutting rules to compute it where τ (k 2 , m 2 ) = 1 − 4m 2 /k 2 .An alternative computation is to derive the discontinuity from the general expression of C 0 (k 2 , m 2 ) for 0 < k 2 < 4m 2 , From the two branches encountered with the ±iϵ prescriptions, the discontinuity is then present only for k 2 > 4m 2 and agrees with Eq. ( 179).The dispersive representation of C 0 (k 2 , m 2 ) in this case is written as with ρ(s, m 2 ) given by Eqs. ( 177) and (179).The equation above allows to reconstruct the scalar integral C 0 (k 2 , m 2 ) from its dispersive part.One can prove the sum rule in (4) by direct integration of ρ.The area under the integral is conserved and in the chiral limit ρ(s, m) → δ(s).

C Chern-Simons terms
Introducing external gauge fields B λ and A µ , the effective action for a chiral anomaly interaction, expressed in terms of such external fields can be modifed by CS terms of the form where (186) Notice that a 1 = a 2 , as expected from the Bose symmetry of the two V lines.It is also well known that the total anomaly a 1 + a 2 + a 3 ≡ a n is regularization scheme independent.We recall that a shift of the momentum in the integrand (p → p + a) where a is the most general momentum written in terms of the two independent external momenta of the triangle diagram (a = α(p 1 + p 2 ) + β(p 1 − p 2 )) induces on ∆ changes that appear only through a dependence on one of the two parameters characterizing a, that is W λµν (β, p 1 , p 2 ) = W λµν (p 1 , p 2 ) − i 4π 2 βϵ λµνσ (p 1σ − p 2σ ) . (187) We have introduced the notation W λµν (β, p 1 , p 2 ) to denote the shifted 3-point function, while W λµν (p 1 , p 2 ) denotes the original one, with a vanishing shift.

Figure 2 :
Figure 2: Distribution of the axial anomaly for the AAA diagram