Renormalised 3-point functions of stress tensors and conserved currents in CFT

We present a complete momentum-space prescription for the renormalisation of tensorial correlators in conformal field theories. Our discussion covers all 3-point functions of stress tensors and conserved currents in arbitrary spacetime dimensions. In dimensions three and four, we give explicit results for the renormalised correlators, the anomalous Ward identities they obey, and the conformal anomalies. For the stress tensor 3-point function in four dimensions, we identify the specific evanescent tensorial structure responsible for the type A Euler anomaly, and show this anomaly has the form of a double copy of the chiral anomaly.


Introduction
The classic results for CFT 2-and 3-point functions are formulated in position space [1][2][3][4]. Nevertheless, for a growing number of modern applications -from conformal anomalies [5][6][7][8][9] to quantum critical transport [10][11][12][13][14][15][16], to holographic cosmology [17][18][19][20][21][22][23][24][25][26][27][28] -it is highly desirable to know the counterpart of these results in momentum space. Our aim in this paper, building on [29], is to fill this gap for tensorial correlators requiring renormalisation. Aside from practical applications, there are a number of other motivations for developing momentum-space CFT. One is the need to better integrate CFT methods with those of QFT more broadly. Away from the critical point, our toolkit is generally limited to perturbative methods such as Feynman diagrams, and such calculations are typically performed in momentum space. Understanding CFT in momentum space then allows us to smoothly interpolate between physics at, and away from, the critical point.
Momentum space is also ideally suited for the study of tensorial correlators, as is our focus here. Constructing a set of basis tensors from the metric and independent momenta, any tensorial correlator can be decomposed into a set of scalar form factors multiplying the elements of this basis. As the trace and transverse Ward identities are algebraic in momentum space, we can moreover eliminate all non-transverse-traceless components by reducing them to simpler lower-point correlators. Only transverse-traceless tensors then appear in our basis, and taking into account the permutation symmetries associated with exchanging identical operators, we quickly arrive at a minimal decomposition for the tensorial structure involving the smallest possible number of form factors.
For 2-and 3-point correlators, these form factors are simply functions of the corresponding momentum magnitudes. The complicated tensorial Ward identities deriving from conformal symmetry then reduce to a simple set of scalar partial differential equations for the form factors. Those corresponding to the special conformal Ward identities factorise and can be solved by elementary separation of variables. The remaining dilatation Ward identities can be solved by performing a Mellin transform to extract the components of appropriate scaling weight. The resulting form factors can be expressed in terms of a class of integrals involving three modified Bessel functions that we call triple-K integrals.
From a practical perspective, this first-principles approach, based on solving the conformal Ward identities directly in momentum space, is considerably more efficient than attempting to Fourier transform the known expressions for position-space correlators. In fact, for the correlators we study in this paper, such an approach fails due to the appearance of divergences arising from integrating over configurations for which the operator insertions coincide. Indeed, the classic position-space expressions for CFT correlators are in any case only valid at non-coincident points: specific contact terms must be added to cancel the divergences in the Fourier transform.
To find the form of these contact terms, and more importantly the momentum-space correlator itself, it is better to return to our first-principles approach. For specific spacetime 1 and operator dimensions, we find the triple-K integrals representing the scalar form factors in our decomposition contain divergences. These divergences can be regulated by infinitesimally shifting the spacetime and operator dimensions. As these are the only dimensionless parameters appearing in the conformal Ward identities, this scheme represents the most general universally-applicable regularisation that preserves conformal invariance. In this scheme the regulated form factors are simply given by triple-K integrals with correspondingly shifted parameters. To extract their divergences, we use the Mellin mapping theorem, which relates the divergences of triple-K integrals to the poles in a series expansion of their integrand. This easy evaluation of divergences is itself another of the advantages of momentum space. Finally, to cancel the divergences, we add suitable local counterterms. For the correlators of stress tensors and conserved currents we study here, these counterterms are constructed purely from the corresponding sources. Their contribution, upon transforming back to position space, corresponds to the missing contact terms in our discussion above. Upon removing the regulator, we then recover the finite, renormalised momentum-space correlator.
The counterterms we add introduce a dependence on the renormalisation scale breaking conformal invariance. The renormalised correlators then obey modified (or 'anomalous') conformal Ward identities containing additional inhomogeneous terms [30] (see also [31] and references therein). As the trace Ward identity is similarly modified, the coefficients appearing in these anomalous Ward identities are exactly those appearing in the trace anomaly.
In the examples we study, it turns out that both the 2-and the 3-point functions are renormalised by the same counterterm. The anomaly coefficient we obtain is then related to the overall normalisation of the 2-point function. Anomalies of this type, corresponding to type B in the classification of [32], represent however only half the picture. In addition, there can be type A anomalies, whose contribution to the trace of the stress tensor vanishes when integrated over all of flat space. Unlike type B anomalies, type A anomalies are therefore scale-invariant: they break special conformal transformations but not dilatations. Type A anomalies arise when the regulated correlator features a diverging coefficient multiplying an evanescent tensor structure that vanishes in the physical spacetime dimension. The result is in fact finite, and does not represent a genuine UV divergence.
Above two dimensions, and assuming parity invariance, the simplest flat-space correlator exhibiting a type A anomaly is the stress tensor 3-point function in four dimensions. Revealing the evanescent structure of the corresponding regulated correlator, and its role in generating the Euler contribution to the trace anomaly, is in fact one of the major motivations for our present study. Setting aside the full problem for later, the general flavour of these ideas can already be appreciated from the 2-point function in two dimensions [32]. This correlator also exhibits a type A anomaly, and the corresponding tensorial structure is naturally far simpler. In fact, the transverse and trace Ward identities alone constrain the momentum-space regulated correlator in d-dimensions to be T µ 1 ν 1 (p)T µ 2 ν 2 (−p) reg = C T T p d Π µ 1 ν 1 µ 2 ν 2 (p), (1.1) momentum conservation has been removed, C T T is a (d-dependent) constant, and Π µνρσ (p) = π µ(ρ (p)π σ)ν (p) − 1 d − 1 π µν (p)π ρσ (p), π µν (p) = δ µν − p µ p ν p 2 (1.2) are the d-dimensional transverse-traceless and transverse projectors respectively. After a moment's consideration, this correlator can be re-written in the form The right-hand side now involves a 3-form and hence vanishes identically in integer dimensions below three, as a spacetime index in the antisymmetrisation must necessarily be repeated. As in d = 2 + 2 dimensions the overall coefficient C T T has a −1 pole, the regulated correlator thus has the 0/0 structure associated with a type A anomaly. 2 For the interested reader, the remaining steps of the renormalisation procedure can be found in appendix A.1. The resulting renormalised correlator, T µ 1 ν 1 (p)T µ 2 ν 2 (−p) = C T T p 2 π µ 1 ν 1 (p)π µ 2 ν 2 (p), (1.4) has a non-vanishing trace associated with anomalous Ward identity, T s = (c/24π)R, where the subscript s indicates an expectation value in the presence of sources and the central charge is given by c = 12πC T T . Through two-dimensional identities [32], we can re-write this trace anomaly as the square of the chiral anomaly: ( µα p α )( νβ p β ). (1.5) Strikingly, as we will see, exactly the same is true for the four-dimensional Euler anomaly! Ultimately, the source of such relations is the 0/0 structure of type A anomalies. While the importance of this mechanism has long been appreciated [32], its operation has not previously been demonstrated for the stress tensor 3-point function due to the greater complexity of the relevant tensorial structure. Our momentum-space approach makes the appearance of this mechanism manifest for the first time.
Earlier related discussions of tensorial CFT correlators in momentum space have appeared in [7,8]. In these works, the momentum-space correlators in four dimensions were obtained through 1-loop Feynman diagram calculations, utilising the observation from position space [2] that a mixture of free conformal scalars, fermions and vectors is sufficient to generate all the tensor structures permitted by conformal symmetry. For general spacetime and operator dimensions, however, this approach is not always available due to the absence of a corresponding free field realisation of the CFT. In such cases, a direct solution of the conformal Ward identities as we develop here appears to be the only available approach. 3 Our plan in this paper then is to solve for the renormalised 3-point correlators of tensorial operators in a general CFT, focusing on correlators of the stress tensor and conserved currents. Our methods extend straightforwardly to mixed correlators involving scalars, as will be discussed in a sequel [33]. The essential ingredients of our approach have been developed over a number of papers, beginning with [29] which introduced our tensorial decomposition for correlators and method for solving the conformal Ward identities. (Related methods for purely scalar correlators were developed independently in [34].) The results of [29] are sufficient to understand all tensorial and scalar correlators for operator and spacetime dimensions in which renormalisation is not required. 4 Our understanding of the divergences arising and the corresponding renormalisation procedure was then developed in [35], focusing on the simple case of purely scalar correlators. Here, we extend this renormalisation procedure to tensorial correlators, which requires a more general regularisation procedure and an understanding of new issues such as tensorial degeneracies that arise in specific spacetime dimensions. A reduction scheme capable of evaluating all the integrals we encounter in this paper may be found in [36].
We begin in section 2 with an extended summary of this method for constructing renormalised tensorial 3-point functions. We review the trace and transverse Ward identities; their role in decomposing the tensorial structure of correlators into scalar form factors; and the classification of conformal Ward identities into primary and secondary. These identities can be solved in terms of triple-K integrals yielding the full momentum-space 3-point functions for generic values of the operator and spacetime dimensions. As we discuss, for specific values of these parameters, divergences are encountered necessitating regularisation and renormalisation.
Our main results for renormalised 3-point functions are then presented in section 3. After reviewing our conventions for 2-point functions, we proceed to analyse correlators of three currents, one stress tensor and two currents, and three stress tensors. The correlator of one current and two stress tensors is trivial (i.e., vanishes up to contact terms) and is omitted from our discussion here; see instead [37] and section 9.10 of [29]. For each correlator, we list the relevant tensorial decomposition, Ward identities and their solution, and tabulate the divergences arising in different spacetime dimensions and their corresponding counterterms. We give explicit results for all regulated form factors valid in arbitrary spacetime dimensions, and carry out the full renormalisation procedure explicitly in dimensions three and four. This enables a precise identification of the anomalies arising and of the anomalous Ward identities satisfied by the renormalised correlators. In section 4, we discuss the type A Euler anomaly arising in the stress tensor 3-point function. This discussion provides the four-dimensional counterpart to our discussion of the two-dimensional type A anomaly above. In section 5 we review how to extract the physical, scheme-independent constants appearing in the renormalised correlators, before concluding in section 6. Four appendices present supplementary material. Appendix A.1 completes our analysis of the two-dimensional anomaly, while A.2 relates alternative definitions of the 3-point function. Appendix A.3 evaluates the counterterm contributions to the stress tensor 3-point function, and appendix A.4 reviews the evanescent tensorial operators arising in three dimensions.

Notations and conventions for momenta
In this paper, we consider CFTs in d ≥ 3 Euclidean dimensions. For simplicity, we will restrict ourselves to the parity-even sector. 5 To get started, in this subsection we begin by recollecting some of our main notations and conventions.
Firstly, to denote correlators with the overall delta function of momentum conservation removed, we employ a double bracket notation, e.g., Due to momentum conservation, only three of the six Lorentz scalars p i · p j for a given 3-point function are independent. To preserve symmetry under permutations of operators, we choose these to be the momentum magnitudes To obtain compact expressions, we define the following symmetric polynomials of the momentum magnitudes, where i, j = 1, 2, 3, as well as the combination By Heron's formula, J is four times the area of the triangle formed by the three momenta. When decomposing tensor structure, in order to preserve symmetry under permutations, we select the independent momenta using a cyclic rule according to the Lorentz index: Here, the numbering of a given Lorentz index derives from the operator insertion it belongs to. Use of this convention is assumed whenever we refer to the coefficient of a particular tensorial structure, e.g., the coefficient of . In other words, before the coefficient is read off, we first replace momenta as required (using momentum conservation) so as to ensure that only momenta consistent with (2.5) appear.

Transverse and trace Ward identities
Let us now recap the origin of the transverse and trace Ward identities satisfied by the correlators. In a renormalised quantum field theory, under a variation of the sources, the variation of the generating functional is where the subscript s on the 1-point functions indicates the presence of nontrivial sources. The gauge field A a µ sources a conserved current J µa associated with a (generally non-Abelian) symmetry group G, where a = 1, . . . , dim G with repeated indices summed. To obtain correlation functions, we functionally differentiate this generating functional with respect to the sources, before restoring them to their background values. (Namely, a flat metric with the gauge field switched off, which we denote by a subscript zero.) We will return to the details of this procedure in section 2.6.
The transverse Ward identities derive from the invariance of the generating functional under gauge transformations and diffeomorphisms. For a gauge field transforming in the adjoint representation, under a gauge transformation α a and diffeomorphism ξ µ , the variation of the sources is where g is the gauge coupling, f abc the structure constant of G, and the generators T a are normalised such that tr(T a T b ) = 1 2 δ ab . Note that α a is a Lorentz scalar so ∇ µ α a = ∂ µ α a . The invariance of the generating functional under each of these transformations then yields the respective Ward identities Functionally differentiating these identities twice with respect to the various sources, we then obtain the transverse Ward identities for 3-point functions. In momentum space, these identities are algebraic and fully determine the longitudinal components of correlators, e.g., (2.11) The trace components of correlators involving the stress tensor are determined by the trace Ward identities. These derive from Weyl transformations, under which where the anomaly A gives the transformation of the renormalised generating functional under a Weyl variation σ. Functionally differentiating twice the generating relation then yields the various 3-point trace Ward identities. We list these identities, and the accompanying transverse Ward identities, in the main results section of the paper. In particular, we will obtain explicit expressions for all the anomalies that arise.

Decomposition of tensor structure
A key advantage of momentum space is that tensorial correlators can be decomposed in a basis of general tensors constructed from the metric and momenta. Each of these tensors appears multiplied by an accompanying scalar form factor, which for 2-and 3-point functions can be written simply as a function of the momentum magnitudes. Moreover, as all trace and longitudinal components of correlators are completely determined by the trace and transverse Ward identities, this basis need only include transverse-traceless tensors. Thus, choosing these tensors so as to respect the permutation symmetries associated with exchanging identical operators, we obtain an extremely efficient representation of tensorial correlators in terms of a minimal number of scalar form factors. Even the most complicated 3-point correlator, that of three stress tensors, can in general be represented in terms of just five scalar form factors. As a full account of this decomposition may be found in [29], here we will limit ourselves to a quick review of the main points. First, we write the transverse and transverse-traceless projectors as π µα (p) = δ µα − p µ p α p 2 , (2.14) It is useful to further introduce a notation for transverse(-traceless) parts of the conserved current and the stress tensor, 6 j µa (p) = π µ α (p)J αa (p), t µν (p) = Π αβ µν (p)T αβ (p). (2.16) Considering, for example, the 3-point function of three currents, the transverse-traceless part then takes the form where X α 1 α 2 α 3 a 1 a 2 a 3 represents the most general third-rank Lorentz tensor that can be constructed from the momenta and the metric, with scalar coefficients depending on the momentum magnitudes and carrying adjoint indices a 1 a 2 a 3 . In fact, for each Lorentz index α j with j = 1, 2, 3, only a single momentum can appear, since the momentum p j is projected out and the remaining two choices are related by momentum conservation: for example, π µ 2 α 2 (p 2 )p α 2 1 = −π µ 2 α 2 (p 2 )p α 2 3 since p 1 = −p 2 −p 3 . This greatly reduces the number of tensor structures that can be written down.
In order to preserve the symmetry under permutations, we choose these independent momenta cyclically according to their Lorentz index following the rule (2.5), i.e., We thus write π µ 2 α 2 (p 2 )p α 2 3 in place of π µ 2 α 2 (p 2 )p α 2 1 , and similarly Π µ Preserving the permutation symmetry in this way leads to a more efficient parametrisation involving fewer form factors, as discussed in section 1 of [29]. We therefore have the decomposition where the two scalar form factors A a 1 a 2 a 3 1 and A a 1 a 2 a 3 2 are general functions of the momentum magnitudes that can be compactly expressed using the symmetric polynomials (2.3). By convention, if no arguments are specified, we assume the standard ordering of momenta . Various symmetry properties of the form factors under permutations can be deduced from the symmetry of the correlator itself, and are listed in the main results section.
For practical purposes, an important feature of this decomposition is that the form factors can easily be read off from a computation of the full 3-point correlator: one does not need to explicitly separate out the projection operators as per (2.17) then manipulate the remainder into the form (2.19). Instead, the form factors can be extracted simply by looking up the coefficients of certain tensor structures in the full correlator. In the present case, for example, one finds in the full correlator J µ 1 a 1 (p 1 )J µ 2 a 2 (p 2 )J µ 3 a 3 (p 3 ) . Again, we list these relations in the main results section. We similarly list the 'reconstruction formulae' that allow the full correlators to be reconstructed from their transverse-traceless parts, using the trace and transverse Ward identities to fill in the missing pieces. These formulae typically involve the operator which arises from decomposing a rank-two tensor into transverse-traceless, longitudinal and trace pieces according to Finally, we note that in specific spacetime dimensions special tensorial degeneracies exist that allow us to reduce the number of independent form factors. For example, the number of independent form factors in the stress tensor 3-point function can be reduced from five to two in three dimensions, and from five to four in four dimensions. We will return to discuss this feature in section 4 and appendix A.4.

Solving the conformal Ward identities
Having decomposed tensorial 3-point functions into scalar form factors, we can now solve for these form factors by imposing the conformal Ward identities. A comprehensive analysis of the momentum-space conformal Ward identities is presented in [29], so here we will once again limit ourselves to a summary of the main points. Inserting the form factor decomposition for a given correlator into the conformal Ward identities, one obtains an individual dilatation Ward identity for each form factor, plus two sets of equations which we refer to as primary and secondary conformal Ward identities [29] (or CWI, for short). The dilatation Ward identities simply require each form factor to be a homogeneous function of the momentum magnitudes with a certain scaling weight, as discussed in section 4.2 of [29]. The primary CWI, which closely resemble the CWI for scalar 3-point functions [35], can be expressed in terms of the second-order differential operators where i, j = 1, 2, 3 and ∆ j denotes the conformal dimension of the j-th operator in a given 3-point function. Thus, in T µ 1 ν 1 J µ 2 J µ 3 for example, we have ∆ 1 = d and ∆ 2 = ∆ 3 = d−1.
Each primary CWI relates K ij acting on a given form factor to a linear combination of other form factors (or else to zero). In total, we obtain two independent primary CWI for each of the form factors present. The primary CWIs can easily be solved through elementary separation of variables, in combination with a Mellin transform to satisfy the dilatation Ward identities [35]. Each form factor is then given by a linear combination of triple-K integrals, defined as where K β j is a modified Bessel function of the second kind. The parameters α and {β j } = {β 1 , β 2 , β 3 } vary according to the case at hand, but generally depend on the spacetime dimension d and the conformal dimensions of operators. In addition, the solution involves a number of arbitrary constants C j multiplying the triple-K integrals, which we refer to as primary constants. To represent the form factors for a given correlator succinctly, we additionally define the 'reduced' triple-K integral where ∆ j is the conformal dimension of the j-th operator, as appropriate to the case at hand, and the {k j } = {k 1 , k 2 , k 3 } are integers. When the triple-K integrals representing a given form factor converge, the solution thus obtained is unique, assuming the absence of collinear singularities [29]. Where triple-K integrals diverge, we instead need to regulate and renormalise, as we will discuss shortly. In such cases, the solution is no longer unique reflecting our freedom to change the renormalisation scheme through the addition of finite counterterms [35]. Where all {β j } indices are half integer, triple-K integrals reduce to elementary integrals and may be trivially evaluated. In many other cases interest, including all those studied here, the triple-K integrals can be evaluated using the reduction scheme presented in [36].
The remaining set of equations derived from the conformal Ward identities, the secondary CWIs, can be expressed in terms of the first-order differential operators Each secondary CWI equates a linear combination of these operators acting on a given form factor to the coefficient of a particular tensor structure appearing in the longitudinal projection of the correlator (i.e., its contraction with a momentum). This coefficient can be read off from the transverse Ward identities. Plugging in our solution for form factors in terms of triple-K integrals, we simply obtain a set of constraints fixing some of the primary constants C j appearing in our original solution. All surviving primary constants are then free parameters characterising the specific CFT at hand.

Regularisation
For specific operator and spacetime dimensions, the triple-K integrals appearing in our solution for form factors diverge. The construction of a suitable regularisation and renormalisation procedure for dealing with these singularities is our main focus in this paper. Correlators for which renormalisation is not required were analysed in [29]. The singularity condition for a general triple-K integral I α{β j } is where n denotes any non-negative integer (i.e., n = 0, 1, 2 . . .) and any independent choice of ± sign can be made for each β j . When this condition is satisfied, the triple-K integral diverges as can be seen by expanding about its lower limit. Whenever this condition is not satisfied, on the other hand, the triple-K integral either converges absolutely, or else diverges but is uniquely defined by analytic continuation in the parameters α, {β j } from regions where it does converge [29,35].
To regulate these divergences, we use the generalised dimensional scheme where u and v j are constants representing a particular choice of scheme (to which we will return below), and the regulator is a small parameter that will ultimately be set to zero after the divergences have been removed through renormalisation. This infinitesimal shift in the spacetime and operator dimensions produces a corresponding shift in the parameters α and {β j } appearing in triple-K integrals. As the singularity condition (2.30) is no longer satisfied, all triple-K integrals and form factors are now finite. Crucially, as d and ∆ j are the only dimensionless parameters appearing in the conformal Ward identities, the scheme (2.31) represents the most general universally-applicable regularisation that preserves conformal invariance. By preserving conformal invariance, we ensure the regulated 3-point functions take a fixed universal form, independent of all details of the theory, which would not be the case had we chosen a regularisation breaking conformal invariance. The form factors of the regulated theory are therefore given by exactly the same set of triple-K integrals we had before, except that all parameters are now shifted in the manner prescribed by (2.31).
Although these triple-K integrals with shifted parameters are finite, they contain poles in which lead to divergences as → 0. In general, these are single poles, although higherorder poles can arise in cases where the singularity condition (2.30) is multiply satisfied through different choices of the ± signs. To proceed, we need to explicitly evaluate the form of these divergences. This can be accomplished very easily as shown in [35], since the divergent poles can be directly read off from a series expansion about x = 0 of the integrand of the regulated triple-K integral. Writing this expansion as whereα andβ j represent the shifted parameters of the regulated integral, the divergences are given by the formula 7 In other words, the divergences arise from terms in the expansion of the form x −1+w (for any finite nonzero w) that become poles in the limit as → 0. In effect, this result follows from the Mellin mapping theorem, which relates singularities in the Mellin transform of a function to its poles. The ease with which divergences can be extracted in this fashion represents an additional advantage of momentum-space methods over position-space techniques such as differential regularisation [46]. 8 Let us now return to discuss the choice of regularisation scheme specified by the parameters u and v j . For conserved currents and stress tensors, as we focus on in this paper, 7 When the singularity condition (2.30) is multiply satisfied, the coefficients c−1+w themselves contain poles in , leading to a higher-order overall divergence. 8 See also [8,47] for an application of differential regularisation to CFT 3-point functions.
gauge and diffeomorphism invariance in fact fix the dimensions of operators. These canonical dimensions need to be preserved in order for the transverse Ward identities to hold in the regulated theory. This requirement leads us to the unique regularisation scheme A subtlety arises, however, which is that for certain correlators this choice of scheme is not always sufficient to regulate all singularities arising in the triple-K integrals for form factors. In such cases, we can start instead from a more general scheme, such as (2.31) with all v j = v for j = 1, 2, 3 but v = u. We then find that triple-K integrals have poles of the form (u − v) −1 that prevent us from naively taking the limit (2.34). Fortunately, this problem is easily circumvented: in the regulated theory, the primary constants multiplying triple-K integrals are in reality functions of the regulating parameters u, v j and . The appearance of, for example, a (u − v) −1 pole in a triple-K integral then requires that the primary constant multiplying this integral has an expansion of the form Equivalently, we could regard this as a re-definition C → (u − v)C . The form factor now has a finite limit as v → u, enabling us to reach the desired scheme (2.34). Applying the secondary conformal Ward identities will now lead to constraints involving the re-defined primary constant C .

Renormalisation
Singularity type. For any divergent triple-K integral, there is a close correspondence between the choice of ± signs appearing in the solution of the singularity condition (2.30), which we refer to as the singularity type, and the locality properties of the divergence [35]. Singularities of type (− − −) (i.e., those satisfying (2.30) with three minus signs) are ultralocal: they are analytic functions of the three squared momenta, corresponding in position space to divergences with support only when all three operator insertions coincide. Singularities of type (− − +) (and its permutations (− + −) and (+ − −)) are instead semilocal: they are analytic in only two of the three squared momenta, corresponding in position space to divergences with with support only when two of the three operator insertions coincide. Singularities of type (−++) (including permutations), and also of type (+++), are instead nonlocal: they are non-analytic in two or more of the squared momenta, and hence in position space are non-vanishing even for separated operator insertions. Note these are nonlocal singularities of individual triple-K integrals, however, and not of the regulated 3-point function itself, which has only ultralocal and/or semilocal singularities. We will return to the interpretation of these singularities shortly. For correlators of three scalars, triple-K integrals of all singularity types can arise. For the correlators we study here, involving only stress tensors and conserved currents, the situation is more straightforward. In fact, in our computations for form factors, we will only encounter triple-K integrals with singularities of type (− − −) and (− − +) (along with permutations), and never singularities of types (− + +) or (+ + +). Moreover, after imposing the secondary conformal Ward identities, we obtain primary constants such that all singularities of type (− − +) actually cancel out when the triple-K integrals are summed to construct the form factors. Thus the regulated form factors have only ultralocal divergences, derived from singularities of type (− − −).
Counterterms. The singularity type of a divergence dictates the manner of its removal 9 . First let us review this procedure in full generality, following [35], before specialising our discussion to correlators of stress tensors and conserved currents. For a general 3-point correlator, we can construct counterterms using either three sources, or else two sources and an operator, along with an appropriate (even) number of covariant derivatives to bring the operator dimension up to the spacetime dimension d. Counterterms with three sources produce ultralocal contributions to the corresponding 3-point function: they can therefore be used exclusively to cancel (− − −) type singularities. Counterterms of this type give rise to conformal anomalies (which are themselves ultralocal functions of the momenta), through their dependence on the renormalisation scale. 10 Counterterms involving two sources and an operator, on the other hand, produce semilocal contributions to 3-point functions. As discussed in [35], their effect is to generate counterterm contributions proportional to the 2-point function of the operator concerned, and thus they can be used exclusively to cancel (− − +) type singularities (along with permutations). Counterterms of this type correspond to a re-definition of the original bare source for the operator: through their dependence on the renormalisation scale, they give rise to a beta function for the new, renormalised source.
Finally, while it is sometimes possible to construct terms of classical dimension d featuring two or more operators, terms of this type should be viewed as multi-trace deformations of the theory. That is, unlike true counterterms, adding such terms changes the theory. 11 Our inability to cancel nonlocal divergences through counterterms means that singularities of types (+ + +) and (− + +) are in effect spurious: they can be removed only by choosing their coefficients (the corresponding primary constants) to vanish as sufficiently high positive powers of the regulator in the limit as → 0. Thus, it is the representation of the 3-point function in terms of triple-K integrals that is singular, and not the 3-point function itself [35]. After these divergences are removed by selecting coefficients vanishing as sufficiently high powers of , one recovers nonlocal 3-point functions satisfying all the usual (non-anomalous) conformal Ward identities. (Note we cannot similarly eliminate singularities of types (− − −) and/or (− − +), however, as this would leave us with ultralocal and/or semilocal 3-point functions, both of which are pure contact terms.) Returning now to the case at hand, for correlators of stress tensors and conserved currents, the only cubic counterterms we can construct are in fact those involving three sources. To preserve the transverse Ward identities, all counterterms must be covariant with respect to gauge transformations and diffeomorphisms, and this in turn requires all source dependence to enter in the form of gauge-invariant field strengths and spacetime curvature invariants. Moreover, in general there are no composite objects transforming as either a metric or a gauge connection, meaning counterterms cannot be constructed by redefining the sources. On purely dimensional grounds then, we cannot construct any local covariant counterterms involving two sources and one operator: while counterterms must have dimension d, both the field strength and the Riemann tensor have dimension two, and the operators T µν and J µa have dimensions d and d − 1 respectively. This absence of counterterms involving two sources and one operator accounts for the observed cancellation of (− − +) type singularities in the regulated form factors.
Instead, the (− − −) type singularities that remain can all be cancelled through covariant counterterms that are cubic in the sources. In any given spacetime dimension, it is a straightforward exercise to construct all such possible counterterms and evaluate their contribution to the 3-point function at hand. After adjusting the counterterm coefficients so as to remove all divergences, the renormalised correlators then follow by sending → 0 to remove the regulator. Due to the addition of counterterms, the trace and conformal Ward identities satisfied by the renormalised correlators are potentially anomalous. The relevant form of these anomalies can easily be determined by inspection in any given case. The conformal anomalies produced in this fashion are the type B anomalies in the terminology of [32]. In addition, there can be type A anomalies which, like the chiral anomaly, are produced through a more subtle 0/0 mechanism; the four-dimensional Euler anomaly we discuss in section 4 is one such example.
Definition of 3-point function. In this paper, we define the 3-point function through functionally differentiating the generating functional three times. In particular, to preserve symmetry under permutations of the operator insertions, we choose to position all the √ g factors outside all functional derivatives, e.g., Notice that with this positioning, the factors of √ g are effectively set to unity. An alternative definition of the 3-point function, which we employed in [29], is to consider three insertions of the operator obtained through a single functional differentiation (i.e., three insertions of the operator corresponding to the usual flat-space stress tensor or conserved current). With this alternative definition, where the functional derivatives reflect the implicit dependence of the operators on the sources. The relation between these two definitions is then By expanding the functional derivatives in a local basis of operators, as described in [29], all correlators involving functional derivatives can be reduced to ordinary 2-point functions.
The difference between these two definitions is thus simply a set of contact terms with support only when two operator insertions coincide. These contact terms are at most semilocal i.e., individually non-analytic in at most a single squared momentum.
Where these contact terms are finite, as was the case for all correlators studied in [29], which of these two definitions of the 3-point function is adopted is purely a matter of choice. One can straightforwardly convert from one definition to the other by adding or subtracting the corresponding contact terms, which can be evaluated by decomposing the functional derivatives in a local basis of operators. The required formulae for this conversion are listed in appendix A.2.
In the present paper, however, we will encounter many cases where these contact terms involving functional derivatives are potentially singular, as the 2-point functions to which they reduce are themselves singular. In view of this, we have adopted here the definition (2.36) based on three functional derivatives. This definition is guaranteed to be finite, unlike that in (2.37), since it corresponds to taking three functional derivatives of the renormalised generating functional with respect to the renormalised sources [50]. Moreover, by subsuming all terms involving functional derivatives into the definition of the 3-point function, a number of steps in our computations are simplified; in particular, the secondary CWIs and reconstruction formulae take a simpler form. Use of the definition (2.36) also facilitates comparison with the literature (e.g., [2]).

Results for renormalised correlators
Having now reviewed the main points of our approach, in this section we present our results for the renormalised 2-and 3-point functions of conserved currents and stress tensors. In each case, we list the tensorial decomposition of the correlator into form factors (including the associated transverse and trace Ward identities and reconstruction formulae), and the solution for the regulated form factors expressed in terms of triple-K integrals. We also enumerate the available counterterms and the conformal anomalies they give rise to, along with the anomalous conformal Ward identities satisfied by the renormalised form factors. Our final results for 3-point functions depend on only a small number of parameters specific to the given CFT at hand. These are the 2-point function normalisation constants, and any primary constants not fixed by the secondary CWIs.

Review of 2-point functions
In our analysis of 3-point functions, we will need various results for 2-point functions. To this end, and to establish the required notation, let us briefly review the renormalisation of CFT 2-point functions. Further details may be found in [29,35].
In spacetimes of general dimension d > 2, the momentum-space 2-point functions are 12 In even dimensions, however, these expressions are unsuitable. In position space, they correspond to local functions with support only when the operator insertions coincide. By adding local counterterms to the action, as below, these 2-point functions can be set to zero, which in a unitary theory would imply the operators vanish identically. In reality, new UV divergences arise as we approach even dimensions d = 2N . In the dimensionally regulated theory, working in a scheme with u = v 1 = v 2 so that operators retain their canonical dimensions, we find The divergences as → 0 can be removed through the addition of suitable counterterms. To quadratic order in the sources, 13 the necessary counterterms are where W µνρσ denotes the Weyl tensor and the renormalisation scale µ appears on dimensional grounds. As we are assuming d ≥ 4, the Laplacians are raised to positive integer powers so these counterterms are local. The counterterm coefficients can be expanded as where the finite pieces c T T represent a particular choice of renormalisation scheme. After removing the regulator by sending → 0, we obtain the renormalised correlators From the perspective of the renormalised theory, however, D JJ and D T T are simply schemedependent constants whose values can be changed arbitrarily by adjusting the renormalisation scale. Unlike (3.1) and (3.2), the renormalised correlators (3.10) and (3.11) are nonlocal, being non-analytic functions of p 2 . Due to the explicit µ-dependence of the counterterms, they acquire an anomalous scale dependence As usual, the anomalous terms on the right-hand side here are finite, local, and scheme independent. They can be derived from the quadratic anomaly action which implies the trace anomaly

General analysis
Decomposition. The transverse Ward identity is The full 3-point function can thus be reconstructed from the transverse-traceless part using This formula is valid in any spacetime dimension d.
Form factors. The tensor decomposition of the transverse-traceless part is The form factors A 1 and A 2 are functions of the momentum magnitudes. If no arguments are given, then we assume the standard ordering, A j = A j (p 1 , p 2 , p 3 ), while by p i ↔ p j we denote the exchange of the two momenta but not the corresponding adjoint indices, e.g., (p 3 , p 2 , p 1 ). The A 1 factor is completely antisymmetric, i.e., for any permutation σ of the set where (−1) σ denotes the sign of the permutation σ. Under a permutation of the momenta only, however, the form factor is completely symmetric, The form factor A 2 is antisymmetric under (p 1 , a 1 ) ↔ (p 2 , a 2 ), i.e., Note that the group structure of the form factors requires the existence of tensors of the form t a 1 a 2 a 3 with appropriate symmetry properties (fully antisymmetric for the one associated with A 1 , and antisymmetric in its first two indices for the one associated with A 2 ). One such tensor is the structure constant f a 1 a 2 a 3 . As argued in [2], the correlation function vanishes if the symmetry group is Abelian.
The form factors correspond to the coefficients of the following tensor structures in the full correlator J µ 1 a 1 ( Here, we assume the independent momenta in the correlator have been chosen according to the index rule (2.5).
Primary CWIs. The primary CWIs are Their solution in terms of triple-K integrals is where the reduced triple-K integrals are defined in (2.25) and C a 1 a 2 a 3 1 and C a 1 a 2 a 3 2 are constants. We refer to all such constants arising in solutions of the primary CWI as primary constants.
Secondary CWIs. The independent secondary CWIs are where the operators L s,n and R s are defined in (2.26) and (2.27). In each of the last lines above, we assume that the independent momenta in the correlator have been selected according to the index rule (2.5). To evaluate these right-hand sides, we apply the transverse Ward identity (3.18). The identity denoted by the asterisk is redundant, i.e., it is trivially satisfied in all cases and does not impose any additional conditions on primary constants.
Divergences. The triple-K integrals appearing in (3.27) and (3.28) have singularities which can be identified by looking for solutions of the singularity condition (2.30). We have compiled the results in table 1 below, using n = 0, 1, 2 . . . to represent any nonnegative integer. The 3-point function is clearly finite in all odd spacetime dimensions and has at most a single pole in for even dimensions d ≥ 4. (To obtain higher-order divergences would require the singularity condition to be multiply satisfied [35].) One can check that the regularisation scheme u = v j for j = 1, 2, 3 is sufficient to regularise all these singularities. As this scheme preserves the canonical dimensions of currents (i.e., ∆ j =d − 1), as required for the transverse Ward identities to hold in the regulated theory, we will adopt it in the following.
Form factor Table 1: Singularities of triple-K integrals in the form factors for J µ 1 J µ 2 J µ 3 .
Counterterms. All relevant counterterms for the parity-even sector are constructed from the field strength F a µν . For 3-point functions, we are interested in terms that are cubic in the source A a µ , which arise from counterterms with two or three F a µν , such as Clearly, the available counterterms are dependent on the spacetime dimension. In four dimensions, only the left-hand counterterm shown is available; crucially, this same counterterm also contributes to the renormalisation of the 2-point function (see (3.6)), so its coefficient is already fixed as given in section 3.1. Above four dimensions, more than one counterterm may be present.

Odd spacetime dimensions
In any spacetime dimension d = 2N for integer N ≥ 1, the secondary CWI leads to where C JJ is the 2-point function normalisation in (3.1). In particular, for d = 2N + 1, this expression simplifies to where we use the convention (−1)!! = 1.
Three dimensions. After re-scaling C a 1 a 2 a 3 1 to remove an overall numerical factor, we find the form factors where the symmetric polynomial a 123 is defined in (2.3).

Even spacetime dimensions
In even dimensions d ≥ 4, both sides of the secondary CWI diverge as −1 . On the lefthand side, the triple-K integrals appearing in the form factors are divergent as discussed above. On the right-hand side, after making use of the transverse Ward identity, the 2point functions that arise are similarly divergent. To handle this, it is useful to expand the primary constants in the regulator as Evaluating the secondary CWIs order by order in then constrains the coefficients C (n) j . A useful trick here, as discussed in section 6.3 of [29], is to focus on the zero-momentum limit in which p 3 → 0 and p 1 , p 2 → p. In this limit, the triple-K integrals reduce to known integrals of only two Bessel functions, yielding 14 To handle derivatives of triple-K integrals with respect to momentum magnitudes, such as arise from acting with the operators L s,n and R s on form factors, we can make use of the recursion relations discussed in section 6.1.2 of [29] (see also section 3.2 of [36]). These relations allow us to re-express derivatives of triple-K integrals in terms of pure triple-K integrals, allowing a straightforward evaluation of the zero-momentum limit.
As it turns out, examination of this limiting momentum configuration is sufficient to evaluate the coefficients C (n) j for general even dimension d = 2N . This general solution reads where H n = n j=1 j −1 is the n-th harmonic number and we use the current-conserving regularisation scheme (2.34). This general solution is valid provided suitable counterterms exist, which as we will see, is always the case.
Four dimensions. Substituting N = 2 in (3.39) and (3.40), we find the regulated solution to the secondary CWIs, The form factor A a 1 a 2 a 3 1 is then finite, while A a 1 a 2 a 3 2 has a single pole, In four dimensions, our only counterterm is (3.6) with N = 2. Moreover, the coefficient of this counterterm has already been determined from the renormalisation of the 2-point function, as given in (3.8). The contribution of this counterterm to the 3-point function must therefore precisely cancel the divergence in the form factor above. Taking three functional derivatives of e −Sct , this counterterm contribution is In terms of form factors, this yields Substituting (3.8), we see this cancels the divergence of the second form factor as required.
The renormalised form factors can now be written where for convenience we have re-labelled the ( -independent) constant C (0)a 1 a 2 a 3 1 as C a 1 a 2 a 3 1 , and D JJ is the scheme-dependent constant defined in (3.10). I (fin) 2{111} represents the finite part of the integral I 2{111} , and is given by (3.48) The integral I 1{000} can be evaluated explicitly in terms of dilogarithms [36], where J 2 is defined in (2.4) and (3.50) Each renormalised form factor A a 1 a 2 a 3 1 and A a 1 a 2 a 3 2 thus contains a nonlocal piece involving triple-K integrals. For A a 1 a 2 a 3 1 this is all we have. For A a 1 a 2 a 3 2 , we have in addition a scaleviolating logarithmic piece containing terms depending explicitly on the renormalisation scale µ, along with a scheme-dependent constant term proportional to D JJ . This constant can be adjusted by changing the renormalisation scale: sending The anomalous scale-dependence of the 3-point function is These expressions also follow from the anomaly action (3.16), which in four dimensions extends to cubic order in A a µ . The same coefficient C JJ thus controls both the 2-and 3-point anomalies.
In the renormalised theory, the primary CWI take the same form as in the regulated theory. The secondary CWI are anomalous, however, and read

General analysis
Decomposition. The transverse and trace Ward identities are As we will see, in odd dimensions anomalies are absent and A µ 2 µ 3 a 2 a 3 can be set to zero. In even dimensions anomalies are generally present; see (3.120) for the case d = 4. Taking the trace of (3.56) and swapping 1 ↔ 2, we see the anomaly is transverse, i.e., p 2µ 2 A µ 2 µ 3 a 2 a 3 = 0 and similarly p 3µ 3 A µ 2 µ 3 a 2 a 3 = 0. The obtain the full 3-point function from the transverse-traceless part, we use the reconstruction formula where T µ 1 ν 1 α is defined in (2.21).
Form factors. The tensorial decomposition of the transverse-traceless part is where all the form factors are functions of the momentum magnitudes A j = A j (p 1 , p 2 , p 3 ).
In the penultimate term, the momenta but not the adjoint indices are exchanged, i.e., while the form factor A 3 does not exhibit any symmetry properties.
Given the 3-point function T µ 1 ν 1 (p 1 )J µ 2 a 2 (p 2 )J µ 3 a 3 (p 3 ) with independent momenta chosen according to (2.5), the form factors can be read off according to Finally, as the anomaly is transverse, we can decompose its tensorial structure as Primary CWIs. The primary CWIs are Their solution in terms of triple-K integrals (2.25) is where the C a 2 a 3 j for j = 1, 2, 3, 4 represent primary constants. In particular, these constants are all symmetric in the group indices, C a 3 a 2 j = C a 2 a 3 j for all j = 1, 2, 3, 4. Consequently, they can be represented by any symmetric bilinear form of the algebra, e.g., the Killing form f a 2 b c f a 3 c b , although other choices may also exist.
Secondary CWIs. The independent secondary CWIs are where L s,n and R s are defined in (2.26) and (2.27). All right-hand sides can be determined using the transverse Ward identities (3.55)-(3.56). The first identity marked by an asterisk is redundant, i.e., it is trivially satisfied in all cases and does not impose any additional constraints on the primary constants.
Divergences. The triple-K integrals in our solution (3.67)-(3.70) have singularities that can be identified using the condition (2.30). (The procedure is similar that for J µ 1 J µ 2 J µ 3 above.) The possible singularities are listed in the following table, where d is the spacetime dimension and n denotes any non-negative integer.
Form factor Regularisation. We cannot immediately work in the desired scheme (2.34), as this fails to regulate the (− − +) singularities in the triple-K integrals for the form factor A a 2 a 3

4
. Instead, we begin in the more general scheme v j = v = u for j = 1, 2, 3, in which all triple-K integrals are regulated, and then redefine the primary constants as necessary in order to obtain a finite limit as v → u.
Counterterms. As there are no counterterms capable of removing (−−+) singularities, 15 in order for the form factor A a 2 a 3 4 to be finite, all the (− − +) singularities must necessarily cancel against one another. The regulated form factors are therefore finite in odd spacetime dimensions, while in even dimensions they contain at most −1 poles, as only the (− − −) singularities survive.
These surviving (− − −) singularities can be removed by counterterms constructed from two gauge-invariant field strengths F a µν and at most a single Riemann tensor, e.g., Higher powers of the Riemann tensor cannot appear as these would generate a vanishing contribution to T µ 1 ν 1 J µ 2 J µ 3 once the metric is restored to flatness. As both the Riemann tensor and the field strength have scaling dimension two, no counterterms are available in odd dimensions, consistent with the finiteness of the form factors. In d = 4 + 2n for any non-negative integer n, we have counterterms involving two field strengths and 2n covariant derivatives, while in d = 6 + 2n we have counterterms involving one Riemann tensor, two field strengths and 2n derivatives. As the counterterms without a Riemann tensor also contribute to the renormalisation of J µ 2 J µ 3 , their coefficients are already fixed by our discussion in section 3.1. In d = 4, all divergences of the 3-point function should then be set by the 2-point normalisation C JJ . The introduction of counterterms gives rise to conformal anomalies as we will discuss.

Odd spacetime dimensions
Our first task is to evaluate the divergences arising in the triple-K integrals for the form factor A a 2 a 3

4
. From table 2, we see that only (− − +) singularities are present, and hence these triple-K integrals have only single poles in . Regulating in the scheme v j = v = u for j = 1, 2, 3, for odd dimensions d = 2N + 1 ≥ 3 we find where H n = n j=1 j −1 denotes the n-th harmonic number and (−1)!! = 1. The divergences in the remaining triple-K integrals can then be obtained using the recursion relations given in section 3.2 of [36], e.g., Assembling the form factor A a 2 a 3 4 according to (3.70), expanding the primary constants as we find that the cancellation of the pole in , as necessitated by the lack of counterterms, requires C . (3.80) We now have to re-define the remaining primary constants so that the v → u limit can be taken, in order to arrive in the scheme (2.34). To this end, it is convenient to set is an undetermined, re-defined primary constant. With this substitution, the limits → 0 and v → u exist and the form factor A a 2 a 3 4 becomes finite. Our next task is to solve the secondary CWI, which are valid in the limit u = v. In fact, it is sufficient to examine only the zero-momentum limit of these equations, making use of (3.37) and (3.38). First, we note the form factors A a 2 a 3 1 , A a 2 a 3 2 and A a 2 a 3 3 are finite and hence do not depend on any of the regulating parameters. As the secondary CWIs (3.71)-(3.73) involve only these form factors, and not A a 2 a 3 4 , we will solve these equations first. Only the leading order 0 terms are relevant, yielding The remaining secondary CWI (3.74) then leads to a dependence between the re-defined primary constant C (1)a 2 a 3 4 and the other primary constants. Its solution is most easily expressed in terms of the constant C (0)a 2 a 3 2 , and reads Together, equations (3.80), (3.81), (3.82) and (3.83) solve the secondary Ward identities, and guarantee that the correlation function is finite in odd spacetime dimensions. While these equations involve two undetermined constants, which we can choose as C

Even spacetime dimensions
The procedure for even spacetime dimensions d = 2N ≥ 4 is similar to that for odd dimensions discussed above. The two main differences are, firstly, the presence of additional (−−−) singularities, which means the triple-K integrals for the form factor A a 2 a 3 4 now have −2 poles. The primary constants must therefore be expanded to order 2 . Secondly, in order to remove the divergences from the regulated form factors, we will need to renormalise by introducing an appropriate counterterm. Evaluating the divergent part of A a 2 a 3 4 in the general scheme v j = v = u for j = 1, 2, 3, we find the singularity as v → u can be removed by re-defining the primary constants as for n = 1, 2. With these substitutions the limit v → u is finite and the regulated secondary Ward identities can be solved. By taking the zero-momentum limit, one arrives at the solution With these primary constants, the limit v → u is finite and the resulting form factors solve the secondary CWI. All that remains is to deal with the singularities in , which are Notice that these surviving singularities are all −1 poles proportional to the 2-point normalisation C JJ . Moreover, there is no dependence on C (0)a 2 a 3 1 , the sole remaining undetermined constant. This fits with our earlier observation that the only available counterterm has been already fixed by the renormalisation of the 2-point function.
Specifically, this counterterm is given by (3.6), setting N = 2. The divergent part of the counterterm coefficient is fixed by (3.8). One can easily verify that the contribution from this counterterm cancels all the divergences in (3.103). We then obtain the renormalised form factors where I , which we have relabelled as C a 2 a 3 1 .
Notice that a change of renormalisation scale µ 2 → e −λ µ 2 is equivalent to sending D JJ → D JJ + λC JJ , both for the 2-point function (3.10) and for the 3-point function.
Each renormalised form factor contains the following pieces. First, there is a nonlocal piece involving contributions derived from triple-K integrals. (For A a 2 a 3 1 this is all we have.) Next, there is a scale-violating logarithmic piece consisting of terms that depend explicitly on the renormalisation scale µ. Finally, there is an ultralocal piece containing terms that are polynomial in the squares of momenta. While the coefficients of certain ultralocal terms involve D JJ and hence are scheme-dependent, one can nevertheless extract schemeindependent information by combining the coefficients of multiple terms. This is a (rare) example of a contact term that is unambiguous. It would be interesting to understand the physics associated with such unambiguous contact terms.
Anomalous Ward identities. The anomalous dilatation Ward identities are while of the primary CWI, only those in the fourth line below develop an anomaly, The anomalous secondary CWIs are The trace anomaly is given by the first term in (3.17), setting N = 2. Expanding to quadratic order in the gauge field, we obtain the anomaly entering the trace Ward identity (3.57) and the reconstruction formula (3.58), (3.120) We can equivalently express this in terms of the form factor decomposition (3.65) as

General analysis
Decomposition. The transverse and trace Ward identities are In odd spacetime dimensions anomalies are absent and A µ 2 ν 2 µ 3 ν 3 can be set to zero. In even dimensions, the anomaly is generally present. Taking the δ µ 2 ν 2 trace of (3.122), swapping 1 ↔ 2, then comparing with the p ν 2 2 contraction of (3.123), we see that the anomaly is transverse, i.e., p ν 2 2 A µ 2 ν 2 µ 3 ν 3 = 0 and similarly p ν 3 3 A µ 2 ν 2 µ 3 ν 3 = 0. To decompose the tensorial structure of the 3-point function, we define the projector where we used (2.21) and (2.22). With the aid of (3.122) and (3.123), we can now write Next, we proceed to expand out the trivial identity Since the 2-point function is transverse and traceless, while the anomaly is transverse (but not traceless), we find that and similarly for permutations. Moreover, (3.131) The full 3-point function can thus be reconstructed from its transverse-traceless part using where the sum is taken over all six permutations σ of the set {1, 2, 3}. The contribution coming from the anomaly is (3.133)

Form factors. The tensorial decomposition of the transverse-traceless part is
The form factors A j , j = 1, . . . , 5 are functions of the momentum magnitudes. If no arguments are specified then the standard ordering is assumed, A j = A j (p 1 , p 2 , p 3 ), while by p i ↔ p j we denote the exchange of the two momenta, e.g., The form factors A 1 and A 5 are symmetric under any permutation of momenta, i.e., for any permutation σ of the set {1, 2, 3}, The remaining form factors are symmetric under p 1 ↔ p 2 , i.e., they satisfy Given the full correlator T µ 1 ν 1 (p 1 )T µ 2 ν 2 (p 2 )T µ 3 ν 3 (p 3 ) with independent momenta chosen according to the index rule (2.5), the form factors can be read off as

140)
In addition to decomposing the transverse-traceless part of the 3-point function, we can introduce an analogous form factor decomposition for the anomaly A µ 2 ν 2 µ 3 ν 3 . Since the anomaly is transverse, but not in general traceless, we first apply (2.22) twice to separate out the transverse-traceless and trace parts. The tensor structure of the transverse-traceless parts can then be decomposed in terms of form factors, yielding Here, the form factors B j = B j (p 1 , p 2 , p 3 ), j = 1, . . . 5 are functions of the momentum magnitudes, with standard ordering assumed unless otherwise specified. Explicit expressions for these form factors in d = 4 are given in (3.216)-(3.220). Using this decomposition, we can re-express the anomalous contribution (3.133) to the reconstruction formula (3.132) as [29] T Primary CWIs. The primary CWIs are (3.144) The solution in terms of triple-K integrals (2.25) is
Secondary CWIs. The independent secondary CWIs are Divergences. The analysis of singularities arising in the solution to the primary CWIs is similar to that presented earlier for J µ 1 J µ 2 J µ 3 . The triple-K integrals in the solution may satisfy the (− − −) and/or various versions of the (− − +) singularity conditions. The following table lists all the possibilities, depending on the spacetime dimension d. As usual, n denotes an arbitrary non-negative integer.

Form factor
Integral Table 3: Singularities of triple-K integrals in the form factors for Regularisation. We cannot immediately work in the desired scheme (2.34), as this fails to regulate the (− − +) singularities in the final two integrals for the form factor A 5 in the table above. Instead, we begin in the more general scheme say v j = v = u for j = 1, 2, 3, in which all triple-K integrals are regulated, and then redefine primary constants as necessary in order to obtain a finite limit as v → u.
Counterterms. As there are no counterterms capable of removing (−−+) singularities 16 , for the form factor A 5 to be finite, all the (− − +) singularities must necessarily cancel 16 See the discussion on page 14.
against one another. The regulated form factors are therefore finite in odd dimensions, while in even dimensions they contain at most −1 poles, as only the (− − −) singularities survive.
To remove these remaining (− − −) singularities, we can add counterterms constructed from the metric in a covariant fashion. Since the counterterms should not contribute to the 1-point function, they must contain either two or three Riemann tensors, e.g., In odd spacetime dimensions, no counterterms are present and hence the 3-point function must be finite and non-anomalous. In even spacetime dimensions, we can construct counterterms with either two or three Riemann tensors, accompanied by an even number of covariant derivatives. The first option requires d = 4 + 2n for non-negative integer n, while the second requires d = 6 + 2n. The introduction of counterterms such as these gives rise to conformal anomalies, as we will discuss.

Odd spacetime dimensions
The form factor A 5 has singularities both as → 0 and as v → u deriving from the integrals J 2{110} and J 0{000} . The former must cancel out, while the latter can be removed by re-defining the corresponding primary constants. To achieve this requires where C (1) 5 represents the undetermined, re-defined primary constant. With these substitutions, the limits v → u and → 0 are now finite.
Next, we solve the secondary Ward identities, which are valid in the limit u = v. Making use of the zero-momentum limit (3.37)-(3.38) to simplify expressions, we obtain the solution  2 , which we have relabelled as C 1 and C 2 . A straightforward check confirms that all primary and secondary CWIs are satisfied.

Even spacetime dimensions
The procedure for even spacetime dimensions d = 2N ≥ 4 is similar to that for odd dimensions discussed above. The appearance of (−−−) singularities leads to the additional complication of −2 divergent triple-K integrals, meaning that primary constants must be expanded to order 2 . In addition, to remove the divergences in regulated form factors we will need to renormalise by introducing counterterms.
Our first task is to remove the singularity as v → u in the form factor A 5 . In addition, as there are no counterterms capable of removing (− − +) singularities, all singularities of this type must cancel among themselves so that only (− − −) singularities remain. The form factor should then have only an −1 divergence in the regulator, meaning the −2 divergences must cancel. These conditions leads to substitutions (3.156) accompanied by a higher-order substitution. In total, With these substitutions, the limit v → u is finite and the regulated secondary CWI can now be solved. By taking the zero-momentum limit, we arrive at the solution Four dimensions. Re-labelling C (0) 1 → C 1 and C (0) 2 → C 2 for convenience, with the substitutions above, the remaining singularities are

177)
To remove these singularities, we have at our disposal the counterterms where the four-dimensional Euler density E 4 and squared Weyl tensor W 2 are given by Together with R 2 , these form a complete basis of quadratic curvature invariants. The contribution from these counterterms to the Weyl variation of the renormalised generating functional is (3.182) In the first line, the variation of the counterterm action can be taken outside the expectation value as it depends only on the sources, and not the dynamical fields. The term proportional to c R in the final line arises as we have chosen in (3.181) to use the four-dimensional square of the Weyl tensor as a counterterm, rather than its d-dimensional counterpart. This choice of scheme has the benefit of simplifying the counterterm contributions. If we now choose the counterterm coefficients we obtain the trace anomaly Since the trace anomaly must be finite, note the R 2 counterterm can only contribute at finite order. In fact, as the b R trace anomaly would generate a nonzero trace for the stress tensor 2-point function, our choice of scheme in section 3.1 requires that b = 0. Moreover, since we have already used the Weyl-squared counterterm to renormalise the stress tensor 2-point function, the coefficient c is already fixed. Comparing (3.179) with (3.7) and (3.9) (or alternatively (3.184) with (3.17)), we find The contribution of the counterterms (3.179) to the transverse traceless form factors can be evaluated as described in appendix A.3. The R 2 counterterm makes no contribution, while the others contribute Despite appearances, however, these are not genuine UV divergences as we will discuss in section 4. From (3.183), we identify the Euler anomaly coefficient Using this relation to eliminate C 2 , we can now express the renormalised form factors as

196)
. (3.197) Here where I (fin) 2{111} and I 1{000} are given in (3.48) and (3.49). The scheme-dependent constant D 1 is related to the finite part of the counterterm coefficients in the regulated theory. In fact, as we will discuss in section 4, D 1 can be removed through a degeneracy of the form factor basis. Note that a change of renormalisation scale µ 2 → e −λ µ 2 is equivalent to sending D T T → D T T − 2λc and D 1 → D 1 + λa; the same applies to the 2-point function (3.11), making use of (3.185).
Each renormalised form factor contains the following pieces. First, there is a nonlocal piece involving contributions derived from triple-K integrals. (For A 1 this is all we have.) Next, there is a scale-violating logarithmic piece consisting of terms that depend explicitly on the renormalisation scale µ. Finally, there is an ultralocal piece containing terms that are polynomial in the squares of momenta. While the coefficients appearing in this ultralocal piece are scheme-dependent, since each involves D T T and/or D 1 , one can nevertheless extract scheme-independent information by combining the coefficients of multiple terms. This is another (rare) instance in which an ultralocal contact term is unambiguous and it would be interesting to understand the physics associated with such terms.
Anomalous Ward identities. The renormalised form factors satisfy the anomalous dilatation Ward identities As we will discuss in section 4, the presence of the Euler coefficient a in these identities is rather surprising and signals the presence of a hidden degeneracy in our form factor basis.
The renormalised form factors satisfy the anomalous primary Ward identities The anomalous secondary Ward identities are Finally, the anomaly A µ 2 ν 2 µ 3 ν 3 appearing in the trace Ward identity (3.123) can be obtained by expanding the trace anomaly (3.184) to quadratic order in the perturbed metric. The result can be expressed in terms of the transverse decomposition (3.142) through the form factors (3.216) These form factors also enter the reconstruction formula (3.132) through the anomalous contribution (3.143). We have included here the terms proportional b purely for completeness: in this paper, b vanishes as otherwise the b R trace anomaly leads to a nonzero trace for the stress tensor 2-point function.

Dimension-dependent degeneracies and the Euler anomaly
The appearance of the Euler coefficient a in the anomalous dilatation Ward identities (3.202)-(3.206) presents us with a puzzle. These equations encode the response of the form factors to a change in the renormalisation scale, or equivalently a dilatation. On the other hand, under a dilatation, the variation of the renormalised generating functional is given by the integral of the trace anomaly. As the integral of the Euler density is a topological invariant, however, all subsequent variations with respect to the metric vanish: the anomalous dilatation Ward identities for the stress tensor 3-point function thus cannot depend on a. All renormalisation scale-dependent terms in correlators must therefore be proportional to the Weyl-squared coefficient c, and not the Euler coefficient a. (In other words, to the coefficient of the type B and not the type A anomaly [32].) How then can the form factors contain scale-dependent terms proportional to a, but not the correlators? This apparent contradiction is resolved by the existence of hidden dimension-dependent degeneracies in our form factor basis. As we will explain in this section, these degeneracies correspond to the existence of nontrivial tensorial structures that vanish in specific spacetime dimensions. When the full tensorial 3-point function is reassembled from its constituent form factors, we find that all the scale-dependent terms proportional to a in fact reassemble into precisely one of these vanishing tensorial structures. Thus, while the individual form factors may contain scale-dependent terms proportional to a, the full tensorial 3-point function does not.
Geometrically, the origin of these dimension-dependent degeneracies is simply the fact that any n-form vanishes in spacetime dimensions d < n. In our present context, these n-forms are constructed from the momenta and metric tensor, although a close analogy exists with the Lovelock identities [51] constructed from spacetime curvature forms. Stripping out all factors of momenta, the underlying identities can also be viewed as Schouten identities [52]. When suitably contracted and projected into a transverse-traceless basis, these vanishing higher forms yield a set of degenerate form factors that produce a vanishing contribution to the full correlator. As these degeneracies involve arbitrary functions, they can even be used to set specific form factors to zero. In three dimensions, as we discuss in appendix A.4, their effect is to reduce the number of independent form factors for the stress tensor 3-point function from five to two. In four dimensions, as we will show here, we can reduce the number of independent form factors from five to four. Our main interest however will be to understand the connection between degeneracies and the Euler anomaly, supplying the four-dimensional counterpart to the two-dimensional analysis of the anomaly in the introduction and appendix A.1.

Dimension-dependent degeneracies
Let us begin with an arbitrary (2,2)-form K α 1 α 2 In any (integer) spacetime dimension d < 5, the totally antisymmetrised product vanishes as an index must necessarily be repeated. Upon contracting indices, we find where, in analogy with the Riemann tensor, we have defined (In fact, choosing K α 1 α 2 β 1 β 2 to be the Riemann tensor yields a class of generalised Lovelock identities, as discussed in [53].) For our present purposes, we will take whereupon Despite appearances, note that K α 1 α 2 β 1 β 2 and its contractions are in fact symmetric under all permutations of the momenta. We can now decompose (4.2) into its component transverse-traceless and trace pieces by contracting with three copies of (2.22). As any contraction of (4.2) with a momentum vanishes, there are no longitudinal pieces. We can therefore write The fully transverse-traceless piece, can be expressed in our standard form factor basis (3.134) as The trace terms can instead be decomposed in the same form as the anomalous contribution to the 3-point function, as given in (3.143) and (3.142). First, from tracing over a single pair of indices, we find along with two similar terms which can be obtained by permutation of index labels and momenta. Next, tracing over two pairs of indices, we obtain along with two similar terms obtainable by permutation. Finally, from tracing over all three pairs of indices, we have Putting everything together, the contributions from (4.9), (4.10) and (4.11) (along with their respective permutations) can be written in precisely the form given in (3.143) and (3.142), with Thus, in summary, we can write where on the right-hand side the first term is transverse traceless and constructed from the form factors (4.8), and the second term takes the form of an anomaly contribution with form factors (4.12). 17 In exactly four dimensions, clearly all the expressions listed in (4.12) vanish due to the explicit factors of (d − 4). As the left-hand side of (4.13) also vanishes in d = 4, the form factors in (4.8) are therefore degenerate: they yield a vanishing contribution to the 3-point function. For any given tensorial component, this can be verified explicitly by writing out all squared momenta in terms of their individual components.

Euler anomaly
Let us now discuss the implications of these results for the stress tensor 3-point function.
Firstly, any multiple of (4.8) that appears in the renormalised form factors can immediately be removed. Studying the form factors in (3.193)- (3.197), we see that the schemedependent terms proportional to D 1 are of precisely this form. The same is also true for the terms proportional to a on the right-hand side of the anomalous dilatation Ward identities (3.202)-(3.206), or equivalently all terms proportional to a ln µ 2 in the the form factors (3.193)- (3.197). As claimed, the scale-dependence of the renormalised correlator thus depends only on the Weyl-squared coefficient c and not on the Euler coefficient a.
More generally, we can remove any multiple of the degenerate form factors in (4.8) by an arbitrary function of the momenta that is symmetric under all permutations. In principle, we could use this freedom to set one of the renormalised form factors to zero, reducing the number of independent form factors from five to four. If one does so, however, the anomalous conformal Ward identities take a more complicated form, and are no longer given by the homogeneous conformal Ward identities plus inhomogeneous terms.
Let us now consider the role of the degeneracy from the standpoint of the regulated theory, where d = 4+2u . Here, the form factors contain divergent −1 poles. If we multiply (4.13) by an overall factor of −4a/u , we find the transverse-traceless form factors Comparing with the actual regulated form factors for the stress tensor 3-point, which we found earlier in (3.174)-(3.178), after using the identification (3.192) we see that all the divergences proportional to a are in fact of exactly this form.
Putting this observation to one side, let us now examine −4a/u times the second term on the right hand side of (4.13). The zeros coming from the factors of (d − 4) in (4.12) are cancelled by this pole, yielding the finite form factors along with vanishing corrections of order . Comparing with the terms proportional to a in the actual anomaly contribution we found earlier in (3.216)-(3.220), we find an exact 17 Note these 'correlators' are simply a short-hand for the corresponding form factor decompositions, rather than representing the actual 3-point correlator.
match up to an overall sign. The form factors (4.15) thus represent minus the Euler anomaly contribution of the 3-point function.
Rearranging (4.13), we see the divergent form factors in (4.14) are equal to the finite Euler anomaly contribution (i.e., minus the form factors in (4.15)) plus −4a/u times the left-hand side of (4.13). This last quantity is of the form 0/0 (namely, an −1 pole multiplying an evanescent tensorial structure that vanishes in d = 4) and is technically finite. In this sense, the form factors in (4.14) do not represent a genuine UV divergence. This behaviour is indeed exactly as we expect for a scale-independent type A anomaly [32]. Any genuine UV divergence (such as that represented by the terms proportional to c) would need to be removed by a divergent counterterm contribution. Such a counterterm contribution would however introduce a non-vanishing dependence on the renormalisation scale, corresponding instead to a scale-dependent type B anomaly.
Evaluating the totally antisymmetrised product in dimensional regularisation is unfortunately an awkward procedure [54], and moreover the result is scheme-dependent. Rather than trying to evaluate −4a/u times the left-hand side of (4.13) explicitly, however, we can instead simply remove this term through the addition of an Euler counterterm. This counterterm generates a (correspondingly finite and scale-independent) contribution of precisely the required form. One may anticipate this results from our discussion in section 3.4.3, where we showed that divergent terms in the form factors which are proportional to a can be cancelled by the Euler counterterm in (3.179). As we will now see the true function of this counterterm is to remove the finite scheme-dependent totally antisymmetrised product.
Computing the metric variation of the Euler counterterm about an arbitrary d-dimensional background, we find Here, to go from the first line to the second, we used and hence where the last term vanishes by the Bianchi identity upon integrating by parts and extracting the final index from the explicit antisymmetrisation.
Setting g µν = δ µν + h µν and expanding to cubic order in the perturbation h µν , we find (4.20) where all indices are now raised with the background metric (i.e., h α β = δ ασ h σβ ). From this expression, we then obtain the counterterm contribution to the 3-point function (4.21) Since a = −a/2u + O( 0 ) (see (3.183)), this counterterm contribution is thus exactly equal to −4a/u times the left-hand side of (4.13). As in our discussion above, it is technically finite meaning all dependence on the renormalisation scale vanishes in the limit → 0. 18 In summary, then, while the regulated form factors (3.174)-(3.178) of the stress tensor 3-point function contain apparent divergences of the form (4.14), these divergent form factors reassemble into a tensorial structure that vanishes in d = 4. The resulting contribution to the regulated 3-point function thus has the 0/0 form expected of type A anomalies, providing an explicit four-dimensional analogue to the two-dimensional discussion of [32]. By adding an Euler counterterm, we can eliminate this 0/0 piece yielding a finite and unambiguous trace anomaly contribution of the expected Euler type.
Finally, tracing over the first pair of indices in (4.21), note that we can write the Euler contribution to the trace anomaly (3.123) as Strikingly, as mentioned in the introduction, this is the "square" of the chiral anomaly. Relaxing parity conservation, the transverse Ward identity (3.18) for the current 3-point function acquires an anomalous term where d a 1 a 2 a 3 is a group theoretic tensor. This chiral anomaly takes the form 24) where N depends on the matter content, and hence we have This relation resembles the double-copy relation between Yang-Mills and gravity scattering amplitudes, see [55] and references therein. It is likely that analogous relations for type A anomalies exist in all higher even dimensions.

Determining the scheme-independent constants
Our efforts thus far have determined the renormalised 3-point functions up to only a few constants. Of these constants, a number are scheme-dependent and can be arbitrarily adjusted by adding finite counterterms or by changing the renormalisation scale. The remainder are scheme-independent, and serve to characterise the particular CFT at hand. These physical, theory-specific constants are: (i) the constant C 1 , which as the only new parameter unrelated to anomalies entering the 3-point function, effectively determines the OPE coefficient; (ii) the normalisation of the relevant 2-point function (either C JJ or C T T ), which also controls the type B anomaly of the 3-point function; and (iii), for the correlator of three stress tensors in four dimensions, the Euler coefficient a parametrising the type A trace anomaly. Our remaining task is thus to extract these physical constants for some given CFT of interest. As C JJ and C T T are already known from the 2-point function, we will focus on the new constants C 1 and a arising in the 3-point function. For perturbative theories, these constants can be identified as follows. First, the scalar Feynman integrals corresponding to the required form factors are found by examining the coefficients of the appropriate tensor structures. We can then either evaluate these integrals directly, in special kinematic configurations for which they simplify, or else map them to triple-K integrals. Here, the relations between Feynman-and/or Schwinger-parametrised integrals and triple-K integrals given in appendix C of [29] are particularly useful.
In the rest of this section, we illustrate both these approaches, using free Dirac fermions in four dimensions as a generic example. The corresponding flat-space operators are thus Note however that on a general curved background, T µν depends on the metric and hence δT µν /δg ρσ is nonzero. Our stress tensor 3-point function then involves a 2-point contribution from this functional derivative operator, according to our definitions (A.10) and (A.17). By decomposing this functional derivative operator in a local basis, as in section 4.3 of [18], one finds however that only the form factor A 5 is affected. As this form factor is not used in our discussion below, the contribution from δT µν /δg ρσ can be safely ignored.

The constant C 1
The constant C 1 is most easily isolated from the form factor A 1 . In the 3-point function, this form factor multiplies the tensorial structure of the highest dilatation weight, namely that built entirely from momenta. In four dimensions, for all the correlators we study, the dilatation weight of A 1 itself is then minus two. As any triple-K integral of negative dilatation weight converges, A 1 is therefore finite and we can evaluate C 1 directly in the bare theory. 19 Let us consider, for example, the case of T µ 1 ν 1 J µ 2 J µ 3 for free fermions in d = 4. The form factor A 1 can be obtained from the 1-loop Feynman-parametrised integral The calculation leading to this result is straightforward and proceeds as discussed in appendix B.3 of [18] and section 7.5 of [29]. 20 Conformal invariance dictates that this parametric integral must however be equivalent to a triple-K integral. Using the mapping given in (C.23) of [29], we can convert the triple-K integral for A 1 (see (3.67)) to a Feynmanparametrised integral, Comparing with (5.2), we can immediately identify Alternatively, we can look for special kinematic configurations in which the relevant integrals simplify. The squeezed (or collinear) limit where one momentum magnitude vanishes is especially useful in this regard. In this limit, triple-K integrals reduce to double-K integrals which can be evaluated via the simple analytic formula 21 For the case at hand, this yields The parametric integral (5.2) also simplifies in this limit, and can be likewise be evaluated: The result may also be obtained directly from the calculations of [7]. Starting from the tensors listed in table 2 of [7], substituting (µ, ν, α, β, k, p, q) → (µ1, ν1, µ2, µ3, −p1, p2, p3) and putting all momenta into their standard form (2.5), we find A1 = −4(F3 + F5 − F7) − 2p 2 2 F9 − 2p 2 3 F10 = −C7 + 2C8 − C9 after using (180)-(182) and (187)-(188). From (178) and table 3 in [7], one then recovers the parametric integral (5.2). 21 The integral converges for Re α > 2|Re β| − 1 but may be extended beyond this range by analytic continuation as discussed in [29].
Using the same methods, we can similarly compute C 1 for all other correlators. For example, for the stress tensor 3-point function of free fermions in d = 4, the form factor A 1 is given by the 1-loop integral This result can be obtained by adapting the analogous three-dimensional calculation in [18]. 22 Comparing with our present result (3.145), using (C.23) of [18] we find and hence C 1 = − 1 5760π 2 . (5.12) Alternatively, we can arrive at this result by evaluating the triple-K and Feynman parametric integrals in the squeezed limit: (5.13)

The Euler coefficient a
While C 1 follows from the form factor A 1 , which is finite in four dimensions, to compute the Euler coefficient a we need to examine the divergences in one (or more) of the regulated form factors A 2 to A 5 . In the appropriate scheme with u = v j for j = 1, 2, 3, combining (3.192) and (3.174)-(3.178), we see these divergences take the form 14)

17)
The form factor A 2 is particularly convenient in that its divergence is independent of the momenta. For free fermions in d = 4, one finds the divergent part of A 2 in the scheme 23 22 The required integral is given by (4.13) in [18], where the loop integral is now four-dimensional rather than three-dimensional. The right-hand side of (4.14) needs to be multiplied by an additional factor of two as the gamma matrices are now four-dimensional, rather than two-dimensional as is the case in d = 3. 23 From appendix C of [29], the scheme u = vj = 1 corresponds to evaluating 1-loop tensorial Feynman integrals in d = 4 + 2 while keeping the powers of momenta δj appearing in the denominators fixed. u = v j = 1 is given by the parametric integral (5.19) and hence a + c = 7 5760π 2 . (5.20) We can now either evaluate the divergence of another form factor, or else simply use the 2point function to compute c via the relation (3.185). Taking this latter route, we calculate for example where the repeated indices are summed over and we used (3.5). We then find in agreement with standard results obtained by other methods [2], [57]. 24 With these values of a and c, the divergences in the remaining form factors A 3 , A 4 and A 5 (including now the contribution from δT µν /δg ρσ ) take the expected form (5.14)-(5.18). Indeed, for additional security, we have checked that we were able to reproduce the full renormalised form factors (3.193)- (3.197). Recovering the correct dependence on the scheme-dependent constant D T T is a particularly nontrivial check, requiring a consistent regularisation for both the 2-and 3-point functions.
To summarise, for perturbative theories all scheme-independent constants entering the renormalised 3-point functions can be obtained through elementary means. The overall normalisation C 1 can be found from the finite form factor A 1 ; either by relating parametrised Feynman integrals to triple-K integrals, or else by directly evaluating the form factor in the squeezed limit. For all correlators apart from that of three stress tensors, the only remaining scheme-independent constants are then the 2-point function normalisations. For the stress tensor 3-point function, one has in addition the Euler coefficient a. As above, this can be evaluated from the divergences of the regulated form factors. Alternatively, if the renormalised correlator is already to hand, we can read off the Euler coefficient from the fully-traced 3-point function: This formula follows from the trace Ward identity (3.123) and the decomposition of the anomaly (3.142) in terms of the form factors (3.216)-(3.220). (We assume here we are working in a scheme where the stress tensor 2-point function is traceless, setting b = 0 in (3.220).) Geometrically, the right-hand side is equal to 64 a times the squared area of the triangle formed from the three momenta.

Discussion
We have solved for the renormalised, tensorial 3-point functions of a general CFT. Our solution highlights the advantages of working in momentum space. First, the tensorial structure of correlators can be decomposed into a minimal set of scalar form factors, which multiply independent basis tensors constructed from the metric and independent momenta. Second, all non-transverse-traceless components of this basis can be eliminated through the trace and transverse Ward identities, whose form in momentum space is algebraic. The resulting scalar form factors now obey simple conformal Ward identities whose form is near-identical to those obtained from purely scalar correlators. Of these momentum-space Ward identities, those corresponding to special conformal transformations factorise and can be solved through elementary separation of variables. The remaining dilatation Ward identities are solved through a Mellin transform leading to triple-K integral solutions. The divergences of these triple-K integrals are readily understood and can be regulated through infinitesimal shifts of the operator and spacetime dimensions. This generalised dimensional regularisation preserves conformal invariance, maintaining our control over the form of the 3-point functions. For correlators of stress tensors and conserved currents, all divergences can then be satisfactorily removed by the addition of counterterms cubic in the sources. (More generally, for mixed 3-point correlators involving scalar operators, other type of counterterms are required as we will discuss in [33].) The resulting renormalised correlators now obey anomalous conformal Ward identities, whose inhomogeneous terms encode the breaking of conformal invariance by the counterterms. In position space, the contribution of these counterterms represents the missing contact terms whose role is to remove the singularities associated with coincident operator insertions.
Besides scheme-dependent terms, only a small number of physical, scheme-independent constants appear in the renormalised 3-point correlators. These are an overall normalisation associated with the form factor of lowest dilatation weight, and the coefficients appearing the trace anomaly. The latter can be split into the coefficients of type B anomalies (which control the 2-point normalisations), and for the correlator of three stress tensors, the Euler coefficient encoding the type A anomaly. The values of all scheme-independent constants can easily be evaluated for any CFT of interest, either by evaluating the form factors in special kinematic configurations, or else by returning them to their canonical representation as triple-K integrals.
As we saw explicitly in d = 4, the scale-independence of the type A anomaly is associated with a UV divergent coefficient multiplying an evanescent operator that vanishes in the physical spacetime dimension. This general structure was predicted long ago in [32], and our present results supply all the remaining details. The geometric origin of these evanescent operators, as forms of higher rank than the spacetime dimension, is particularly clear in momentum space. Interestingly, the Euler contribution to the anomaly of the stress tensor 3-point function takes the form of the square of the chiral anomaly. The same also holds in two dimensions and it is likely to be true in all even dimension. It would be interesting to understand if there is a deeper meaning to this relation, or if it simply follows from kinematics. Even in the latter case, it may still have non-trivial implications, for example by allowing results established for chiral anomalies (e.g. descent relations) to be applied to (the Euler part of) conformal anomalies.
Looking ahead, we hope our momentum-space methods will open new approaches; not only to contemporary problems, such as bootstrapping tensorial correlators [58], but to classic problems whose past investigation has been hindered by the complications of tensorial structure. In particular, our improved understanding of the stress tensor 3-point function could provide renewed impetus for the following investigations: Nonlocal effective actions. As 2-and 3-point functions are universal, there should exist a nonlocal geometric effective action, whose quadratic and cubic parts are universal, reproducing the full stress tensor 2-and 3-point functions. What is the form of this effective action in four dimensions? While the 3-point contribution from the Euler anomaly can be obtained from the Riegert action [9,[59][60][61], doubts remain about the physicality of this action [3,32,62,63]. Moreover, this anomalous contribution represents just one part of the full correlator [64]. The situation in four dimensions is thus quite different to that in two, where, since all two-dimensional metrics are conformal to flat space, only the anomalous contribution is present; the Polyakov action obtained by integrating the anomaly then captures all the information in stress tensor correlators. In four dimensions, the form of the nonlocal effective action reproducing the full 2-and 3-point functions remains unknown.
Anomaly matching. The matching of conformal anomalies between broken and unbroken phases, proposed by Schwimmer and Theisen in [65], is a fundamental ingredient in the proof of the a-theorem [66,67]. The physical importance of anomaly matching demands that all the arguments of [65] should be put on as explicit a footing as possible. Now that we have the full stress tensor 3-point function to hand, it should be possible to clarify certain remaining ambiguities. What are the precise tensor structures corresponding to the A and B amplitudes of [65]? Can we verify that the analytic structure of the B amplitude, at the special kinematic point where all Lorentz invariants vanish, degenerates from a branch cut to an apparent pole? How does this enable us to isolate the Euler anomaly, and how precise an analogy can we make with the chiral anomaly [68,69] ?
A spectral proof of the a-theorem? In two dimensions, an elegant proof of Zamolodchikov's c-theorem [70] can be obtained from the positivity of the Källén-Lehmann spectral representation [71]. Is there an analogous spectral proof for the four-dimensional atheorem? 25 Through standard dispersion relations, we can construct spectral representations for the form factors of the stress tensor 3-point function [5]. These spectral functions, corresponding to the imaginary parts of form factors, then obey homogeneous (i.e., nonanomalous) conformal Ward identities, as all counterterm contributions are analytic in the squared momenta and thus are projected out. A successful spectral approach to the a-theorem could provide complementary insight to the anomaly matching and dilaton effective action arguments of [66,67]. The simplicity of our tensorial decomposition -involving only four form factors after removal of the degeneracy -could yield new insight.
Defining an a-function. As we have seen, at fixed points of RG flow we can extract the Euler anomaly coefficient a from various projections of the renormalised 3-point function: from the fully traced correlator as in (5.23), for example, but also from the transverse traceless part. Projections such as these are natural candidates for an a-function: they are well-defined observables, away from the fixed point, that reduce to the Euler coefficient at the fixed point itself. It would be interesting to investigate whether any of these candidates also exhibit the required monotonicity under RG flow.
As we see, exactly the same 3-form appears here as in the regulated 2-point function (1.3). Thus, in d = 2 + 2 , even if the counterterm coefficient c has a −1 divergence as → 0, the vanishing of the 3-form in two dimensions means that this counterterm contribution is finite. In addition, the dependence on the renormalisation scale drops out.
To connect the form of this counterterm contribution with that of the regulated 2-point function, we now need to separate out its transverse-traceless and trace pieces. This can be accomplished using two copies of the projection operator (2.22). As the counterterm contribution is purely transverse (i.e., vanishes when any free index is contracted with the momentum), we can write this projection using only the transverse-traceless and transverse projectors introduced in (1.2). We thus have 6(d − 1) p 2 π µ 1 ν 1 (p)π µ 2 ν 2 (p) . (A.4) The first transverse-traceless term now matches the form of the regulated 2-point function.
The second term has a nonzero trace, and crucially an extra factor of (d − 2) which arises from the tracing over the 3-form: This extra factor of (d − 2) ensures the second term in (A.4) is finite as → 0, even though the corresponding tensor structure is nonvanishing. If we choose the counterterm coefficient as minus that of the regulated 2-point function, the transverse-traceless first term of the counterterm contribution (A.4) now cancels the regulated 2-point function (1.3). In the limit → 0, we obtain T µ 1 ν 1 (p)T µ 2 ν 2 (−p) = C T T p 2 π µ 1 ν 1 (p)π µ 2 ν 2 (p). (A.7) This renormalised correlator is independent of the renormalisation scale, as we expect for a type A anomaly. Let us emphasise once again that both the counterterm contribution and the regulated 2-point function are each individually finite in the limit as → 0. Our introduction of the counterterm is simply a convenient device for evaluating this limit: rather than evaluating the antisymmetrisations in d = 2 + 2 , which is rather awkward [54], we simply cancel them against one another. Instead of introducing this counterterm, an alternative way to define the limit → 0 is simply to replace all tensorial structures with their d = 2 equivalents, while keeping all scalar coefficients in d = 2 + 2 . This procedure is discussed for the 2-point function in [32]. One could take a similar approach to understand the Euler anomaly for the stress tensor 3-point function in four dimensions. In section 4, however, we find it easier to proceed as discussed above, by introducing a finite counterterm to remove the antisymmetrised term.

A.2 Relating 3-point function definitions
Here we collect for easy reference the formulae needed to convert the 3-point functions appearing in our previous work [29] to those of the present paper. As discussed on page 15, in this paper we define the 3-point functions by Those used in [29] are instead defined through where the functional derivatives reflect the implicit dependence of the operators on the sources. As 2-point functions of operators with different dimensions vanish in a CFT, these terms can only appear when the corresponding dimensions match.

A.3 Evaluation of counterterm contributions
In this appendix, we evaluate the counterterm contributions to the form factors for the stress tensor 3-point function in four dimensions. As these form factors are associated with the transverse-traceless part of the correlator, it suffices to work in a gauge where the inverse metric perturbation is transverse traceless, 26 g µν = δ µν + γ µν , γ µµ = 0, γ µν,ν = 0, The Ricci curvature   26 In this section, all raised indices should be understood as being raised with the full perturbed metric.
Repeated lowered indices should be summed using the flat background metric. Commas denote partial derivatives and ∂ 2 = ∂µ∂µ. Our conventions follow those of [72].
We now find the Ricci-squared counterterm, As expected, the Euler counterterm generates no contribution at quadratic order in γ µν . Note its variation can also be computed as described in section 4; projecting (4.21) into a transverse traceless basis leads to the same form factors as we find here.
It is interesting to consider the three-dimensional degeneracy (A.45) from a geometrical perspective. Unlike in four-dimensions, where the degeneracy (4.8) derives from the existence of an evanescent counterterm, in three dimensions there are no counterterms. Instead, the identity (A.45) is analogous to the vanishing of the Weyl tensor: replacing K α 1 α 2 β 1 β 2 in (A.45) by the Riemann curvature R α 1 α 2 β 1 β 2 , the right-hand side is equal to the Weyl tensor W α 1 α 2 β 1 β 2 , which vanishes in three dimensions by precisely this identity.