1 Introduction
Abstract

We present a study of the correlation function of three stress-energy tensors in d dimensions using free field theory realizations, and compare them to the exact solutions of their conformal Ward identities (CWI’s) obtained by a general approach in momentum space. The identification of the corresponding form factors is performed within a reconstruction method, based on the identification of the transverse traceless components (A_{i}) of the same correlator. The solutions of the primary CWI’ s are found by exploiting the universality of the Fuchsian indices of the conformal operators and a re-arrangement of the corresponding inhomogenous hypergeometric systems. We confirm the number of constants in the solution of the primary CWI’s of previous analysis. In our comparison with perturbation theory, we discuss scalar, fermion and spin 1 exchanges at 1-loop in dimensional regularization. Explicit checks in d=3 and d=5 prove the consistency of this correspondence. By matching the 3 constants of the CFT solution with the 3 free field theory sectors available in d=4, the general solutions of the conformal constraints is expressed just in terms of ordinary scalar 2- and 3-point functions (B_{0},C_{0}). We show how the renormalized d=4 TTT vertex separates naturally into the sum of a traceless and an anomaly part, the latter determined by the anomaly functional and generated by the renormalization of the correlator in dimensional regularization. The result confirms the emergence of anomaly poles and effective massless exchanges as a specific signature of conformal anomalies in momentum space, directly connected to the renormalization of the corresponding gravitational vertices, generalizing the behaviour found for the TJJ vertex in previous works.

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The General 3-Graviton Vertex (TTT) of Conformal Field Theories

in Momentum Space in d=4

Claudio Corianò and Matteo Maria Maglio

Dipartimento di Matematica e Fisica "Ennio De Giorgi"

Università del Salento

and

INFN Lecce,

Via Arnesano, 73100 Lecce, Italy

1 Introduction

Exact results in four dimensional conformal field theories (CFT’s) have gathered a lot of attention along the years, mostly because the enlarged SO(2,4) symmetry of such theories has been essential for determining the structure of the correlators, especially for 2- and 3-point functions. These are derived by imposing on them the corresponding conformal Ward identities (CWI’s), which in even spacetime dimensions are broken by the conformal anomaly [1].
Such analysis have traditionally been performed in coordinate space, where the conformal constraints are readily implemented. We recall that the equations for a certain correlator are first solved for separate external spacetime coordinates - giving the homogeneous, conformally invariant solutions - and, at a second stage, the contribution due to the conformal anomaly is taken into account by adding an ultralocal term [2, 3], which is generated when all the external coordinate points coalesce. In dimensional regularization (DR) in d=4 this is achieved by the standard E and C^{2} counterterms, corresponding to the Euler-Poincare’ density and to the Weyl tensor squared respectively. The anomaly is generated by the d-dimensional trace of such counterterms, which is responsible for the appearance of a finite inhomogeneous term as d\to 4. One of the goal of our analysis will be to describe step by step the perturbative renormalization for the TTT, illustrating the separation between the anomaly (or trace) and the traceless parts in the free field theory realization of such correlator, which may turn useful for future studies of the conformal anomaly action. As we are going to discuss, we will work in full generality, and in d=3 and 4 our analysis matches the most general CFT solution, providing the simplest realization of such correlator. We anticipate that our results are in complete agreement with previous analysis of similar correlators such as the TJJ [4, 5, 6]. In particular, this works extends the perturbative results contained in [7], where some of the methods used for the transition to momentum space have been extensively discussed and to which we refer for further details.

1.1 The hierarchy of the CWI’s and the BMS reconstruction

The conformal constraints are hierarchical, with 3-point functions defined in terms of 2-point functions, and so on for higher orders. For 3-point functions they are strong enough to fix the solutions modulo few constants. For 4-point functions the solutions are only partially fixed, due to the presence of an arbitrary function of two conformally invariant ratios. Completely traced correlators of 4-T’s 6-T’s and higher are special, since they are fixed by the first 4 correlators of the same type, due to the presence of conformal trace relations [8], given the quartic nature of the Wess-Zumino action, as derived by Weyl gauging [9] or by other methods.
Undoubtedly, the theoretical interest in this family of correlators is remarkable because of the appearance, beside of the conformal, also of mixed anomalies with axial vector currents [10, 11, 12, 13, 14] [15, 16, 17].
In coordinate space, the solution of the conformal constraints [2] has been obtained for several correlation functions, among which the most demanding one was the TTT, i.e. the correlator of three stress-energy tensors (with free indices). Semi-local contributions, which are generated by a partial pinching of the external coordinates, appear on the right hand side of the conformal WI’s, multiplied by 2-point functions, showing that a complete reconstruction of tensor correlators is indeed possible in a hierarchical way. In this process, conservation and trace WI’s are determined by correlators of lower orders (e.g 2-point functions in the TTT case).

As shown by Osborn and Petkou in coordinate space [2] and by Bzowski, McFadden and Skenderis (BMS approach) [18] in momentum space, the solutions of such equations are determined up to few constants, which characterize a specific CFT. BMS have shown that it is possible to formulate a complete reconstruction procedure for tensor correlators which extends the case of scalar correlators [19]. In the case of a Lagrangian realization of a CFT, such constants are determined by the field content, i.e. the number of scalars, fermions etc. appearing in the theory, and for a sufficient number of independent family sectors and particle multiplicities they are expected to saturate the exact (non Lagrangian) solution and provide its simplest realization.
Such perturbative solutions, obviously, remain valid beyond perturbation theory and match the general CFT prediction for 3-point functions in specific cases, as we are going to show. The d=3 and d=4 cases are such, since the presence of two free field theories sectors, scalars and fermions in d=3, and of scalars, fermions and vectors in d=4, for instance, allow to perform a matching between general and Lagrangian solutions in the most general way.
The simplifications obtained from the perturbative analysis, respect to the general solution - no matter if expressed in terms of 3K integrals (i.e. integrals of 3 Bessel functions)[18], or, as we are going to show, directly in terms of the Appell’s function F_{4} [19, 7], discussed in [7] and here in Section B - are remarkable. In fact, in the latter case, we are allowed to use only B_{0} and C_{0}, the scalar 2- and 3-point functions of ordinary perturbation theory at one loop, to express the full result. Obtaining such expressions for the TTT or for other vertices and using a specific reconstruction method which keeps the number of form factors minimal, allows to achieve a large simplification of the entire approach. At the end of our analysis we will summarize the explicit way to obtain the reconstruction of such a vertex, which may turn useful also for further studies in quantum gravity.

1.2 The role of perturbation theory in the equivalence

While the equivalence between the CFT and the free field theory solutions is obviously expected at some level, the search for an exact match between the two approaches not only provides a simplification of the results but also offers some physical intuition about the origin of the conformal anomaly, once we move to momentum space. In coordinate space this dynamical interpretation is simply absent. In fact, conformal and chiral anomalies show a striking similarity, in momentum space, due to the emergence of anomaly poles [4, 5, 20, 21]. The general nature of this mechanism is easily understood in momentum space by dispersion theory, and unifies chiral and conformal anomalies, as evident from the anomaly supermultiplet [22]. They provide a possible unifying tract of both, which otherwise appear to be completely unrelated. A simple physical description of this phenomenon is indeed currently possible in perturbation theory (see the discussion in our companion paper [7]), at least in the TJJ and in its supersymmetric version.
It is associated, in the integration over the loop momentum (k), to the region of k describing the decay of a massive external graviton line into two collinear massless particles, and then turning into two photons (t-channel cut) [7]. The result of this interaction is in the generation of a term (\sim\beta(g)/k^{2}), proportional to the \beta-function of a theory, which vanishes if the theory is conformal at quantum level (\beta=0). We have recently shown that this phenomenon, in the TJJ case, holds beyond perturbation theory [7] having compared exact solutions with their Lagrangian realization in QED and it is associated to renormalization.

The pole structure describes in the light-cone behaviour of a theory and of its anomaly in terms of its intrinsic degrees of freedom, without the introduction of a Goldstone mode, such as in the case of the Wess-Zumino action. It is more than an educated guess to predict that in the infrared such an action is expected to take a second form. This would be a spontaneously broken version of the same theory, with an asymptotic degree of freedom in the form of a dilaton, the Goldstone mode of broken conformal simmetry, and with the emergence of 3- and 4- dilaton interactions in the infrared, as described by its Wess-Zumino form [23, 8, 24]. For such a reason, the possibility of matching the CFT and the perturbative solutions of the CWI’s in momentum space for the more complex case of the TTT can help to clarify these issues, at least up to cubic level in the fluctuations of the external gravitational metric. At the same time such analysis is a mandatory initial step for further studies of the exact form of the conformal anomaly action.

1.3 Perturbative matchings

The perturbative matching between the general and the Lagrangian solutions of the TTT can be obtained by investigating a certain number of independent sectors for such correlator in general (d) spacetime dimensions. Given the fact that the general solution of the CWI’s depends on 2 independent constants in d=3, and on 3 for d\geq 4 [2], the study of this correspondence is performed, in perturbation theory, by the inclusion of 2 sectors (fermion and scalar) for odd values of d, with the addition of a gauge sector in d=4, which is conformal invariant only in d=4. Compared with the d=3 and d=5 cases, where the TTT is finite, in d=4 the Feynman diagrams need to be renormalized, with the generation of an anomalous contribution, and the matching with the general CFT solution, in this case, is complete.
We should also mention that in higher even dimensions the use of antisymmetric forms running in the loops, which take the same role of the spin 1 sector of d=4, may allow to extend our analysis, providing a third independent sector. In odd spacetime dimensions only the case d=3 is entirely matched, since in this case it has been shown that only two constants are necessary in order to characterize the general solution of the CWI’s. The analysis presented in [2] in coordinate space and in [25, 26] in momentum space agree on the presence of 3 independent constants in the general solution for d>3, which clearly cannot be completely matched by the two conformally invariant free field sectors (scalars and fermions) which are available in odd dimensions. The results for the TTT presented in [25] for d=3 and d=5 are rather simple, and are in agreement with the result obtained by us by combining the two free field theory sectors (scalars and fermions) which are available in the same dimensions. In d=5, for instance, the solution given in [25] is accurately matched, in our case, by such sectors, but it is expected to correspond to a particular solution of the conformal constraints. It is therefore possible that the additional CFT’s described by the general solution d=2k+1 correspond to interacting CFT’s which do not find a perturbative free-field theory realization.

1.4 Reconstruction

The explicit expression of the TTT vertex in momentum space is expected to be very involved, unless one is able to identify a specific procedure in order to reduce the number of independent form factors and bring them into correspondence with the available solution in coordinate space. Attempts in this direction have been made in the past [27, 28], but a general method that introduces a minimal set of form factors which does the job and allows to reconstruct the entire correlator has been proposed only more recently for the TTT,TTO,TJJ correlators [25]. The method is based on a reconstruction program for such correlators which starts from their transverse traceless sectors and builds up the entire correlator exploiting the endomorphic action of the special conformal transformation, with the longitudinal components determined with the use of the conservation and trace Ward identities. The method is completely autonomous compared to coordinate space [29] and allows to derive scalar equations for the transverse traceless form factors which are then solved in terms of 3K integrals. An alternative approach, which bypasses such integrals has been presented by us for the TJJ in our companion work [7], which is based on the observation that the Fuchsian indices of the conformal Ward identities are universal for such systems of equations [7]. We will show how to re-arrange the hypergeometric differential equations of the CWI’s in such a way to generate their non-homogenous solutions starting from the homogenous ones, extending our method to from the TJJ to the TTT. This parts of our analysis is quite independent of the rest of the work but it confirms that the set of the primary CWI’s is indeed determined by a set of 5 constants, in agreement with [25]. This provides a second independent check on the number of constants present in such solutions before the imposition of the constraints of momentum conservation (secondary Ward identities).

As we are going to show in a forthcoming work, our approach can be extended in order to look for special solutions for more general correlators, of 4- and higher points, which are, obviously, not completely fixed by the underlying conformal symmetry. We hope to come back to this point in the future.

1.5 Perturbative solutions

One of the main issues with the general solution is that it gets very involved in the presence of divergences, and requires an entirely new regularization procedure for such 3K integrals, which, however, does not make transparent the fact that the result has to be clearly equivalent to the perturbative one. We have not attempted to compare our results with those of [29] for d=4, but we have verified their complete agreement in d=3 and 5 using our d-dimensional computation. By the same token, the anomalous CWI’s will be derived using our Lagrangian framework and are as general as those derived in [29], but in a far more direct and simplified form.
The study of the matching between the general and the perturbative solutions, and the check of their equivalence, will be done by working in d=3 and 5 dimensions, in order to prove the consistency of our results with the general solution obtained from CFT [25]. The matching to perturbation theory brings in significant simplification of the general solution in terms of 3K integrals, or the very same solution in terms of Appell’s hypergeometrics that we wil present below.
Notice that due to the need or regularizing the solution, 3K integrals [25, 30, 29, 26] are not the master integrals of perturbation theory, since the propagators appearing in the loop - after a suitable conversion - do not carry integers exponents. They cannot be handled by the ordinary reduction procedures which are typical of the multiloop analysis in QCD, due to the need of shifting the exponents in the Feynman propagators of the integrands by a (real) regulator. This has motivated us to reconsider independently all the BMS reconstruction [29] from a simple perturbative perspective. While this follows overall the original proposal, some of the relations concerning the projected special CWI’s have been reobtained using an independent strategy. For instance, we have made an extensive use of Lorentz Ward identities, not mentioned in the original work, in order to come to a final agreement with the expressions quoted in [29].

2 The TTT and TTO correlators

We start by stating our definitions and conventions. We introduce the ordinary definition of the energy-momentum tensor in terms of the generating functional of the theory \mathcal{W} in the Euclidean case

 \braket{T^{\mu\nu}(x)}=\mbox{\small$\displaystyle\frac{2}{\sqrt{g(x)}}$}\mbox{% \small$\displaystyle\frac{\delta\mathcal{W}}{\delta g_{\mu\nu}(x)}$} (2.1)

where

 \mathcal{W}=\mbox{\small$\displaystyle\frac{1}{\mathcal{N}}$}\int\,\mathcal{D}% \,\Phi\ e^{-S} (2.2)

with \mathcal{N} a normalization factor. \Phi denotes all the quantum fields of the theory and S is the quantum action. For the multi-graviton vertices, it is convenient to define the corresponding correlation function as the n-th functional variation with respect to the metric of the generating functional \mathcal{W} evaluated in the flat-space limit

 \begin{split}\displaystyle\braket{T^{\mu_{1}\nu_{1}}(x_{1})\dots T^{\mu_{n}\nu% _{n}}(x_{n})}\equiv&\displaystyle\left[\mbox{\small$\displaystyle\frac{2}{% \sqrt{g(x_{1})}}$}\dots\mbox{\small$\displaystyle\frac{2}{\sqrt{-g(x_{n})}}$}% \mbox{\small$\displaystyle\frac{\delta^{n}\mathcal{W}}{\delta g_{\mu_{1}\nu_{1% }}(x_{1})\dots\delta g_{\mu_{n}\nu_{n}}(x_{n})}$}\right]_{flat}\\ \displaystyle=&\displaystyle\left.2^{n}\mbox{\small$\displaystyle\frac{\delta^% {n}\mathcal{W}}{\delta g_{\mu_{1}\nu_{1}}(x_{1})\dots\delta g_{\mu_{n}\nu_{n}}% (x_{n})}$}\right|_{flat}\end{split} (2.3)

so that it is explicitly symmetric with respect to the exchange of the metric tensors. The 3-point function we are interested in studying is found through (2.3) for n=3

 \begin{split}&\displaystyle\braket{T^{\mu_{1}\nu_{1}}(x_{1})T^{\mu_{2}\nu_{2}}% (x_{2})T^{\mu_{3}\nu_{3}}(x_{3})}=8\,\bigg{\{}-\Braket{\mbox{\small$% \displaystyle\frac{\delta S}{\delta g_{\mu_{1}\nu_{1}}(x_{1})}$}\,\mbox{\small% $\displaystyle\frac{\delta S}{\delta g_{\mu_{2}\nu_{2}}(x_{2})}$}\,\mbox{% \small$\displaystyle\frac{\delta S}{\delta g_{\mu_{3}\nu_{3}}(x_{3})}$}}\\ &\displaystyle\hskip 42.679134pt+\Braket{\mbox{\small$\displaystyle\frac{% \delta^{2}S}{\delta g_{\mu_{1}\nu_{1}}(x_{1})\delta g_{\mu_{2}\nu_{2}}(x_{2})}%$}\,\mbox{\small$\displaystyle\frac{\delta S}{\delta g_{\mu_{3}\nu_{3}}(x_{3})% }$}}+\Braket{\mbox{\small$\displaystyle\frac{\delta^{2}S}{\delta g_{\mu_{1}\nu% _{1}}(x_{1})\delta g_{\mu_{3}\nu_{3}}(x_{3})}$}\,\mbox{\small$\displaystyle% \frac{\delta S}{\delta g_{\mu_{2}\nu_{2}}(x_{2})}$}}\\ &\displaystyle\hskip 42.679134pt+\Braket{\mbox{\small$\displaystyle\frac{% \delta^{2}S}{\delta g_{\mu_{2}\nu_{2}}(x_{2})\delta g_{\mu_{3}\nu_{3}}(x_{3})}%$}\,\mbox{\small$\displaystyle\frac{\delta S}{\delta g_{\mu_{1}\nu_{1}}(x_{1})% }$}}-\Braket{\mbox{\small$\displaystyle\frac{\delta^{3}S}{\delta g_{\mu_{1}\nu% _{1}}(x_{1})\delta g_{\mu_{2}\nu_{2}}(x_{2})\delta g_{\mu_{3}\nu_{3}}(x_{3})}$% }}\bigg{\}}\end{split} (2.4)

where the angle brackets denote the vacuum expectation value. The last term is identically zero in DR, being proportional to a massless tadpole. The first term on the rhs of (2.4) has the diagrammatic representation of a triangle topology, while the contribution of a second functional derivative times a single derivative of the action is interpreted in the perturbative analysis as a bubble diagram. We will keep in mind such decompositon which will be relevant in the last papert of our work once we come to the perturbative analysis .

3 Canonical and trace Ward Identities

The conformal constraints for the TTT correspond to dilatation and special conformal transformations, beside the usual Lorentz symmetries. Generically

 \sum_{j=1}^{3}G_{g}(x_{j})\braket{T^{\mu_{1}\nu_{1}}(x_{1})\,T^{\mu_{2}\nu_{2}% }(x_{2})\,T^{\mu_{3}\nu_{3}}(x_{3})}=0, (3.1)

where G_{g} are the generators of the infinitesimal symmetry transformations. Among these, the conservation WI in flat space of the stress-energy tensor can be obtained by requiring the invariance of \mathcal{W}[g] under diffeomorphisms of the background metric

 \mathcal{W}[g]=\mathcal{W}[g^{\prime}] (3.2)

where g^{\prime} is the transformed metric under the general infinitesimal coordinate transformation x^{\mu}\to x^{\prime\mu}=x^{\mu}+\epsilon^{\mu}

 \delta g_{\mu\nu}=\nabla_{\mu}\epsilon_{\nu}+\nabla_{\nu}\epsilon_{\mu}. (3.3)

It generates the relation

 \nabla_{\nu}\braket{T^{\mu\nu}}=0 (3.4)

while naive scale invariance gives the traceless condition

 g_{\mu\nu}\braket{T^{\mu\nu}}=0. (3.5)

These have been the only constraints taken into account in previous perturbative studies of the TJJ [5, 4, 21] and TTT [28]. The functional differentiation of (3.4) and (3.5) allows to derive ordinary Ward identities for the various correlators. For the three point function case these take the form

 \displaystyle\partial_{\nu}\langle T^{\mu\nu}(x_{1})T^{\rho\sigma}(x_{2})T^{% \alpha\beta}(x_{3})\rangle \displaystyle=\bigg{[}\langle T^{\rho\sigma}(x_{1})T^{\alpha\beta}(x_{3})% \rangle\partial^{\mu}\delta(x_{1},x_{2})+\langle T^{\alpha\beta}(x_{1})T^{\rho% \sigma}(x_{2})\rangle\partial^{\mu}\delta(x_{1},x_{3})\bigg{]} \displaystyle\quad-\bigg{[}\delta^{\mu\rho}\langle T^{\nu\sigma}(x_{1})T^{% \alpha\beta}(x_{3})\rangle+\delta^{\mu\sigma}\langle T^{\nu\rho}(x_{1})T^{% \alpha\beta}(x_{3})\rangle\bigg{]}\partial_{\nu}\delta(x_{1},x_{2}) \displaystyle\quad-\bigg{[}\delta^{\mu\alpha}\langle T^{\nu\beta}(x_{1})T^{% \rho\sigma}(x_{2})\rangle+\delta^{\mu\beta}\langle T^{\nu\alpha}(x_{1})T^{\rho% \sigma}(x_{2})\rangle\bigg{]}\partial_{\nu}\delta(x_{1},x_{3})\,. (3.6)

In order to move to momentum space we fix some conventions. The Fourier transform of the correlators is defined as

 \displaystyle\braket{T^{\mu_{1}\nu_{1}}(p_{1})\,T^{\mu_{2}\nu_{2}}(p_{2})\,T^{% \mu_{3}\nu_{3}}(p_{3})}=\int d^{d}x_{1}d^{d}x_{2}d^{d}x_{3}e^{i\left(p_{1}% \cdot x_{1}+p_{2}\cdot x_{2}+p_{3}\cdot x_{3}\right)}\braket{T^{\mu_{1}\nu_{1}% }(x_{1})\,T^{\mu_{2}\nu_{2}}(x_{2})\,T^{\mu_{3}\nu_{3}}(x_{3})} (3.7)

and similarly for the 2-point function. Translational invariance introduces an overall \delta(P) with P being the sum of all the (incoming) momenta, with the generation of derivative terms \delta^{\prime}(P), after the action of the special conformal transformations on the integrand. Such terms can be investigated rigorously using the theory of tempered distributions, formulated using a symmetric basis. The analysis has been presented in [7] for a Gaussian basis, to which we refer for more details. In our conventions, we have chosen p_{3} as the dependent momentum {p_{3}}\to-p_{1}-p_{2}. Eq. (3) becomes

 \displaystyle p_{1\nu_{1}}\braket{T^{\mu_{1}\nu_{1}}(p_{1})\,T^{\mu_{2}\nu_{2}% }(p_{2})\,T^{\mu_{3}\nu_{3}}({p_{3}})} \displaystyle=-p_{2}^{\mu_{1}}\braket{T^{\mu_{2}\nu_{2}}(p_{1}+p_{2})T^{\mu_{3% }\nu_{3}}({p_{3}})}-{p_{3}^{\mu_{1}}}\braket{T^{\mu_{2}\nu_{2}}(p_{2})T^{\mu_{% 3}\nu_{3}}(p_{1}+{p_{3}})} \displaystyle+p_{2\alpha}\left[\delta^{\mu_{1}\nu_{2}}\braket{T^{\mu_{2}\alpha% }(p_{1}+p_{2})T^{\mu_{3}\nu_{3}}({p_{3}})}+\delta^{\mu_{1}\mu_{2}}\braket{T^{% \nu_{2}\alpha}(p_{1}+p_{2})T^{\mu_{3}\nu_{3}}({p_{3}})}\right] \displaystyle+{p_{3\alpha}}\left[\delta^{\mu_{1}\nu_{3}}\braket{T^{\mu_{3}% \alpha}(p_{1}+{p_{3}})T^{\mu_{2}\nu_{2}}(p_{2})}+\delta^{\mu_{1}\mu_{3}}% \braket{T^{\nu_{3}\alpha}(p_{1}+{p_{3}})T^{\mu_{2}\nu_{2}}(p_{2})}\right]. (3.8)

In the next section, in order to clarify that differentiation in p_{3} has to be performed with the chain rule, we will denote with \bar{p}_{3}^{\mu}\equiv-p_{1}^{\mu}-p_{2}^{\mu}, the dependent momentum, and the independent 4-momenta will be p_{1}^{\mu} and p_{2}^{\mu}. Concerning the naive identity (3.5), it generates the non-anomalous condition

 g_{\mu_{1}\nu_{1}}\braket{T^{\mu_{1}\nu_{1}}(p_{1})\,T^{\mu_{2}\nu_{2}}(p_{2})% \,T^{\mu_{3}\nu_{3}}(p_{3})}=0 (3.9)

valid in the d\neq 4 case.
After renormalization this equation is modified by the contribution of the conformal anomaly, given by the general expression

 \displaystyle g_{\mu\nu}(z)\langle T^{\mu\nu}(z)\rangle \displaystyle= \displaystyle\sum_{I=F,S,G}n_{I}\bigg{[}\beta_{a}(I)\,C^{2}(z)+\beta_{b}(I)\,E% (z)\bigg{]}+\frac{\kappa}{4}n_{G}F^{a\,\mu\nu}\,F^{a}_{\mu\nu}(z) (3.10) \displaystyle\equiv \displaystyle\mathcal{A}(z,g)\,,

by considering only the scheme independent terms with

 \displaystyle\beta_{a}(S) \displaystyle=-\frac{3\pi^{2}}{720}\,,\hskip 28.452756pt\beta_{b}(S)=\frac{\pi% ^{2}}{720}\,, \displaystyle\beta_{a}(F) \displaystyle=-\frac{9\pi^{2}}{360}\,,\hskip 28.452756pt\beta_{b}(F)=\frac{11% \pi^{2}}{720} \displaystyle\beta_{a}(G) \displaystyle=-\frac{18\pi^{2}}{360}\,,\hskip 28.452756pt\beta_{b}(G)=\frac{31% \pi^{2}}{360} (3.11)

being the contributions to the \beta functions coming from scalars (S), fermions (F) and vectors (G). We have defined the two tensors

 \displaystyle C^{2} \displaystyle=R_{abcd}R^{abcd}-\mbox{\small$\displaystyle\frac{4}{d-2}$}R_{ab}% R^{ab}+\mbox{\small$\displaystyle\frac{2}{(d-2)(d-1)}$}R^{2}, \displaystyle E=R_{abcd}R^{abcd}-4R_{ab}R^{ab}+R^{2} (3.12)

being the square of the Weyl conformal tensor and the Euler-Poincare’ density respectively, while R_{abcd} is the Riemann curvature tensor and R_{ab} and R are the Ricci tensor and the Ricci scalar, respectively. Then we get the anomalous WI

 \displaystyle g_{\mu_{1}\nu_{1}}\braket{T^{\mu_{1}\nu_{1}}(p_{1})T^{\mu_{2}\nu% _{2}}(p_{2})T^{\mu_{3}\nu_{3}}(p_{3})} \displaystyle                 =4\,\mathcal{A}^{\mu_{2}\nu_{2}\mu_{3}\nu_{3}}(p% _{2},p_{3})-2\,\braket{T^{\mu_{2}\nu_{2}}(p_{1}+p_{2})T^{\mu_{3}\nu_{3}}(p_{3}% )}-2\,\braket{T^{\mu_{2}\nu_{2}}(p_{2})T^{\mu_{3}\nu_{3}}(p_{1}+p_{3})} \displaystyle                 =4\,\bigg{[}\beta_{a}\,\big{[}C^{2}\big{]}^{\mu_% {2}\nu_{2}\mu_{3}\nu_{3}}(p_{2},p_{3})+\beta_{b}\,\big{[}E\big{]}^{\mu_{2}\nu_% {2}\mu_{3}\nu_{3}}(p_{2},p_{3})\bigg{]} \displaystyle                          -2\,\braket{T^{\mu_{2}\nu_{2}}(p_{1}+p_% {2})T^{\mu_{3}\nu_{3}}(p_{3})}-2\,\braket{T^{\mu_{2}\nu_{2}}(p_{2})T^{\mu_{3}% \nu_{3}}(p_{1}+p_{3})}. (3.13)

We just remark that the solutions of all the conformal constraints, in this study, are obtained by working with the non-anomalous expressions of the corresponding CWI’s, while the anomaly contributions, as in (3.13), are obtained only after taking the d\to 4 limit of the general solution and the inclusion of the corresponding counterterms. All these points will be investigated rather thoroughly in the following sections. We briefly pause to comment on the relation between the current and previous analysis [28] of the TTT in free field theory. The expression for the TTT given in [28] has been presented in a complete form only for the gravitational amplitude g_{1}(p_{1})\to g_{2}(p_{2})+g_{3}(p_{3}), with g_{2} and g_{3} on-shell gravitons, which is quite involved. The expression given in [28] breaks the full symmetry of the correlator and requires a basis of 13 form factors, which is nonminimal. A symmetric and manageable reconstruction of this vertex requires a complete reanalysis of the correlator, with the inclusion also of the special conformal and dilatation constraints, which lower the number of independent form factors to a minimal number, which will be 5. This is the step that we are going to undertake starting from the next section.

4 Special conformal and dilatation WI’s

Dilatation and special conformal WI’s in position space can be derived in various ways, and the transition to momentum space can be made rigorous by taking suitable distributional limits of the derivative of the Dirac delta functions, as discussed by us in [7]. In coordinate space, for the TTT, the special CWI’s take the form

 \displaystyle 0={K}^{\kappa}\braket{T^{\mu_{1}\nu_{1}}(x_{1})T^{\mu_{2}\nu_{2}% }(x_{2})T^{\mu_{3}\nu_{3}}(x_{3})} \displaystyle=\sum_{i=1}^{3}{K_{i}}^{\kappa}_{scalar}(x_{i})\braket{T^{\mu_{1}% \nu_{1}}(x_{1})T^{\mu_{2}\nu_{2}}(x_{2})T^{\mu_{3}\nu_{3}}(x_{3})} \displaystyle+2\left(\delta^{\mu_{1}\kappa}x_{1\rho}-\delta_{\rho}^{\kappa}x_{% 1}^{\mu_{1}}\right)\braket{T^{\rho\nu_{1}}(x_{1})T^{\mu_{2}\nu_{2}}(x_{2})T^{% \mu_{3}\nu_{3}}(x_{3})}+2\left(\delta^{\nu_{1}\kappa}x_{1\rho}-\delta_{\rho}^{% \kappa}x_{1}^{\nu_{1}}\right)\braket{T^{\mu_{1}\rho}(x_{1})T^{\mu_{2}\nu_{2}}(% x_{2})T^{\mu_{3}\nu_{3}}(x_{3})} \displaystyle+2\left(\delta^{\mu_{2}\kappa}x_{2\rho}-\delta_{\rho}^{\kappa}x_{% 2}^{\mu_{2}}\right)\braket{T^{\mu_{1}\nu_{1}}(x_{1})T^{\rho\nu_{2}}(x_{2})T^{% \mu_{3}\nu_{3}}(x_{3})}+2\left(\delta^{\nu_{2}\kappa}x_{2\rho}-\delta_{\rho}^{% \kappa}x_{2}^{\nu_{2}}\right)\braket{T^{\mu_{1}\nu_{1}}(x_{1})T^{\mu_{2}\rho}(% x_{2})T^{\mu_{3}\nu_{3}}(x_{3})} \displaystyle+2\left(\delta^{\mu_{3}\kappa}x_{3\rho}-\delta_{\rho}^{\kappa}x_{% 3}^{\mu_{3}}\right)\braket{T^{\mu_{1}\nu_{1}}(x_{1})T^{\mu_{2}\nu_{2}}(x_{2})T% ^{\rho\nu_{3}}(x_{3})}+2\left(\delta^{\nu_{3}\kappa}x_{3\rho}-\delta_{\rho}^{% \kappa}x_{3}^{\nu_{3}}\right)\braket{T^{\mu_{1}\nu_{1}}(x_{1})T^{\mu_{2}\nu_{2% }}(x_{2})T^{\mu_{3}\rho}(x_{3})} (4.1)

written in terms of a scalar contribution

 {{K}_{i}}^{\kappa}_{scalar}=-x_{i}^{2}\frac{\partial}{\partial x_{\kappa}}+2x_% {i}^{\kappa}x_{i}^{\tau}\frac{\partial}{\partial x_{i}^{\tau}}+2\Delta_{i}x_{i% }^{\kappa} (4.2)

and of spin parts. In momentum space this becomes with \Delta_{i}, i=1,2,3 being the scaling dimensions of 3 generic rank-2 operators - here fixed to be d for the T^{\mu\nu} -

 \displaystyle\sum_{j=1}^{2}\left[2(\Delta_{j}-d)\mbox{\small$\displaystyle% \frac{\partial}{\partial p_{j}^{\kappa}}$}-2p_{j}^{\alpha}\mbox{\small$% \displaystyle\frac{\partial}{\partial p_{j}^{\alpha}}$}\mbox{\small$% \displaystyle\frac{\partial}{\partial p_{j}^{\kappa}}$}+(p_{j})_{\kappa}\mbox{% \small$\displaystyle\frac{\partial}{\partial p_{j}^{\alpha}}$}\mbox{\small$% \displaystyle\frac{\partial}{\partial p_{j\alpha}}$}\right]\braket{T^{\mu_{1}% \nu_{1}}(p_{1})\,T^{\mu_{2}\nu_{2}}(p_{2})\,T^{\mu_{3}\nu_{3}}(\bar{p}_{3})} \displaystyle\qquad+2\left(\delta^{\kappa(\mu_{1}}\mbox{\small$\displaystyle% \frac{\partial}{\partial p_{1}^{\alpha_{1}}}$}-\delta^{\kappa}_{\alpha_{1}}% \delta^{\lambda(\mu_{1}}\mbox{\small$\displaystyle\frac{\partial}{\partial p_{% 1}^{\lambda}}$}\right)\braket{T^{\nu_{1})\alpha_{1}}(p_{1})\,T^{\mu_{2}\nu_{2}% }(p_{2})\,T^{\mu_{3}\nu_{3}}(\bar{p}_{3})} \displaystyle\qquad+2\left(\delta^{\kappa(\mu_{2}}\mbox{\small$\displaystyle% \frac{\partial}{\partial p_{2}^{\alpha_{2}}}$}-\delta^{\kappa}_{\alpha_{2}}% \delta^{\lambda(\mu_{2}}\mbox{\small$\displaystyle\frac{\partial}{\partial p_{% 2}^{\lambda}}$}\right)\braket{T^{\nu_{2})\alpha_{2}}(p_{2})\,T^{\mu_{3}\nu_{3}% }(\bar{p}_{3})\,T^{\mu_{1}\nu_{1}}(p_{1})}=0. (4.3)

Notice that the spin part acts only on two of the three tensors, in this case T^{\mu_{1}\nu_{1}}(p_{1}) and T^{\mu_{2}\nu_{2}}(p_{2}), leaving T^{\mu_{3}\nu_{3}} as a spin singlet [7]. Notice that the Leibnitz rule for the action of the conformal operator K^{\kappa} is violated and the differentiation respect to the third momentum is performed implicitly. The final result shown above, as explictly discussed in [7], is a consequence of the Lorentz WI, which has to be used quite extensively. This takes the form

 \sum_{j=1}^{3}L_{\mu\nu}(x_{j})\langle T^{\mu_{1}\nu_{1}}(x_{1})T^{\mu_{2}\nu_% {2}}(x_{2})T^{\mu_{3}\nu_{3}}(x_{3})\rangle=0 (4.4)

with

 L_{\mu\nu}(x)=\left(i(x_{\mu}\partial_{\nu}-x_{\nu}\partial_{\mu})+\bar{\Sigma% }_{\mu\nu}\right) (4.5)

being the generators of the symmetry, separated into the angular momentum component and in the spin part, with \bar{\Sigma} being the spin generators of SO(4) in the vector representation

 \left(\bar{\Sigma}_{\rho\sigma}\right)_{\mu\alpha}=i\left(\delta_{\rho\mu}% \delta_{\sigma\alpha}-\delta_{\rho\alpha}\delta_{\sigma\mu}\right). (4.6)

In the case of the TTT this gives

 \displaystyle 0 \displaystyle=\sum_{j=1}^{3}L_{\mu\nu}(x_{j})\braket{T^{\mu_{1}\nu_{1}}(x_{1})% T^{\mu_{2}\nu_{2}}(x_{2})T^{\mu_{3}\nu_{3}}(x_{3})} \displaystyle=\sum_{j=1}^{3}i\left(x_{j}^{\mu}\frac{\partial}{\partial{x_{j}}_% {\nu}}-x_{j}^{\nu}\frac{\partial}{\partial{x_{j}}_{\mu}}\right)\braket{T^{\mu_% {1}\nu_{1}}T^{\mu_{2}\nu_{2}}T^{\mu_{3}\nu_{3}}}+2(\bar{\Sigma}^{\mu\nu})^{(% \mu_{1}}_{\alpha_{1}}\langle T^{\nu_{1})\alpha_{1}}T^{\mu_{2}\nu_{2}}T^{\mu_{3% }\nu_{3}}\rangle \displaystyle                 +2(\bar{\Sigma}^{\mu\nu})^{(\mu_{2}}_{\alpha_{2}% }\langle T^{\mu_{1}\nu_{1}}T^{\nu_{2})\alpha_{2}}T^{\mu_{3}\nu_{3}}\rangle+2(% \bar{\Sigma}^{\mu\nu})^{(\mu_{3}}_{\alpha_{3}}\braket{T^{\mu_{1}\nu_{1}}T^{\mu% _{2}\nu_{2}}T^{\nu_{3})\alpha_{3}}} (4.7)

and takes the form in momentum space

 \displaystyle\sum_{j=1}^{2}\left[p_{j}^{\nu}\mbox{\small$\displaystyle\frac{% \partial}{\partial p_{j\mu}}$}-p_{j}^{\mu}\mbox{\small$\displaystyle\frac{% \partial}{\partial p_{j\nu}}$}\right]\braket{T^{\mu_{1}\nu_{1}}(p_{1})\,T^{\mu% _{2}\nu_{2}}(p_{2})\,T^{\mu_{3}\nu_{3}}(\bar{p}_{3})} \displaystyle\qquad+2\left(\delta^{\nu}_{\alpha_{1}}\delta^{\mu(\mu_{1}}-% \delta^{\mu}_{\alpha_{1}}\delta^{\nu(\mu_{1}}\right)\braket{T^{\nu_{1})\alpha_% {1}}(p_{1})\,T^{\mu_{2}\nu_{2}}(p_{2})\,T^{\mu_{3}\nu_{3}}(\bar{p}_{3})} \displaystyle\qquad+2\left(\delta^{\nu}_{\alpha_{2}}\delta^{\mu(\mu_{2}}-% \delta^{\mu}_{\alpha_{2}}\delta^{\nu(\mu_{2}}\right)\braket{T^{\nu_{2})\alpha_% {2}}(p_{2})\,T^{\mu_{3}\nu_{3}}(\bar{p}_{3})\,T^{\mu_{1}\nu_{1}}(p_{1})} \displaystyle\qquad+2\left(\delta^{\nu}_{\alpha_{3}}\delta^{\mu(\mu_{3}}-% \delta^{\mu}_{\alpha_{3}}\delta^{\nu(\mu_{3}}\right)\braket{T^{\nu_{3})\alpha_% {3}}(\bar{p}_{3})\,T^{\mu_{1}\nu_{1}}(p_{1})\,T^{\mu_{2}\nu_{2}}(p_{2})}=0. (4.8)

Similarly, the dilatation WI in coordinate space

 \sum_{j=1}^{n}\left(i\,x_{j}^{\alpha}\frac{\partial}{\partial x_{j}^{\alpha}}+% \Delta_{j}\right)\braket{T^{\mu_{1}\nu_{1}}(x_{1})\,T^{\mu_{2}\nu_{2}}(x_{2})% \,T^{\mu_{3}\nu_{3}}(x_{3})}=0. (4.9)

can be rewritten in momentum space in the form

 \displaystyle\left[\sum_{j=1}^{3}\Delta_{j}-2d-\sum_{j=1}^{2}\,p_{j}^{\alpha}% \mbox{\small$\displaystyle\frac{\partial}{\partial p_{j}^{\alpha}}$}\right]% \braket{T^{\mu_{1}\nu_{1}}(p_{1})\,T^{\mu_{2}\nu_{2}}(p_{2})\,T^{\mu_{3}\nu_{3% }}(\bar{p}_{3})}=0. (4.10)

5 Reconstruction in the BMS approach

In this section we are going to review the reconstruction method of [25] with the inclusion of extra derivations and details specific to the TTT case, which may illustrate more clearly its formulation. The basic idea of the approach is to introduce a symmetric decomposition of the correlator in terms of its transverse traceless and longitudinal sectors. A second ingredient is that the second order differential equations (primary WI’s) which act on the corresponding form factors separate from the first order ones coming from the conservation Ward identities (secondady WI’s).
For this one needs the transverse, transverse-traceless and longitudinal projectors

 \displaystyle\pi^{\mu}_{\alpha} \displaystyle=\delta^{\mu}_{\alpha}-\frac{p^{\mu}p_{\alpha}}{p^{2}},\qquad% \tilde{\pi}^{\mu}_{\alpha}=\frac{1}{d-1}\pi^{\mu}_{\alpha} \displaystyle\Pi^{\mu\nu}_{\alpha\beta} \displaystyle=\frac{1}{2}\left(\pi^{\mu}_{\alpha}\pi^{\nu}_{\beta}+\pi^{\mu}_{% \beta}\pi^{\nu}_{\alpha}\right)-\frac{1}{d-1}\pi^{\mu\nu}\pi_{\alpha\beta}, \displaystyle\mathcal{I}^{\mu\nu}_{\alpha} \displaystyle=\frac{1}{p^{2}}\left[2p^{(\mu}\delta^{\nu)}_{\alpha}-\frac{p_{% \alpha}}{d-1}(\delta^{\mu\nu}+(d-2)\frac{p^{\mu}p^{\nu}}{p^{2}})\right] \displaystyle\mathcal{I}^{\mu\nu}_{\alpha\beta} \displaystyle=\mathcal{I}^{\mu\nu}_{\alpha}p_{\beta}=\frac{p_{\beta}}{p^{2}}% \left(p^{\mu}\delta^{\nu}_{\alpha}+p^{\nu}\delta^{\mu}_{\alpha}\right)-\frac{p% _{\alpha}p_{\beta}}{p^{2}}\left(\delta^{\mu\nu}+(d-2)\frac{p^{\mu}p^{\nu}}{p^{% 2}}\right) \displaystyle\mathcal{L}^{\mu\nu}_{\alpha\beta} \displaystyle=\frac{1}{2}\left(\mathcal{I}^{\mu\nu}_{\alpha\beta}+\mathcal{I}^% {\mu\nu}_{\beta\alpha}\right)\qquad\tau^{\mu\nu}_{\alpha\beta}=\tilde{\pi}^{% \mu\nu}\delta_{\alpha\beta} (5.1)
 \displaystyle\delta^{\mu\nu}_{\alpha\beta} \displaystyle=\Pi^{\mu\nu}_{\alpha\beta}+\Sigma^{\mu\nu}_{\alpha\beta} \displaystyle\Sigma^{\mu\nu}_{\alpha\beta} \displaystyle\equiv\mathcal{L}^{\mu\nu}_{\alpha\beta}+\tau^{\mu\nu}_{\alpha% \beta}. (5.2)

The previous identities allows to decompose a symmetric tensor into its transverse traceless (via \Pi), longitudinal (via \mathcal{L}) and trace parts (via \tau), or on the sum of the combined longitudinal and trace contributions (via \Sigma). Each insertion of stress energy tensor is separated into its longitudinal, transverse traceless and trace parts, in the notation of [25]

 T^{\mu\nu}=t^{\mu\nu}+t_{loc}^{\mu\nu} (5.3)

with

 \displaystyle t_{loc}^{\mu\nu}(p) \displaystyle=\frac{p^{\mu}}{p^{2}}Q^{\nu}+\frac{p^{\nu}}{p^{2}}Q^{\mu}-\frac{% p^{\mu}p^{\nu}}{p^{4}}Q+\frac{\pi^{\mu\nu}}{d-1}(T-\frac{Q}{p^{2}}) \displaystyle=\Sigma^{\mu\nu}_{\alpha\beta}T^{\alpha\beta} (5.4)

and

 t_{loc}^{\mu\nu}=\mathcal{I}^{\mu\nu}_{\alpha}Q^{\alpha}+\frac{\pi^{\mu\nu}}{d% -1}T. (5.6)

We turn to the case of the the 3-graviton vertex. By acting with these projectors on the TTT, the 3-point function is divided into two parts: the transverse-traceless part and the local part (indicated by subscript loc) expressible through the transverse and trace Ward Identities. We will be using the suffix "i" in K_{i},\pi_{i},\Pi_{i} to indicate operators of momentum p_{i}. In the notation of [25], the transverse traceless contributions are denoted as

 \braket{t^{\mu_{1}\nu_{1}}(p_{1})t^{\mu_{2}\nu_{2}}(p_{2})t^{\mu_{3}\nu_{3}}(p% _{3})}={\Pi_{1}}^{\mu_{1}\nu_{1}}_{\alpha_{1}\beta_{1}}{\Pi_{2}}^{\mu_{2}\nu_{% 2}}_{\alpha_{2}\beta_{2}}{\Pi_{3}}^{\mu_{3}\nu_{3}}_{\alpha_{3}\beta_{3}}% \braket{T^{\alpha_{1}\beta_{1}}(p_{1})T^{\alpha_{2}\beta_{2}}(p_{2})T^{\alpha_% {3}\beta_{3}}(p_{3})} (5.7)

while the local contributions, defined by either longitudinal or trace projections, are indicated as

 \displaystyle\langle t^{\mu_{1}\nu_{1}}_{loc}(p_{1})T^{\mu_{2}\nu_{2}}(p_{2})T% ^{\mu_{3}\nu_{3}}(p_{3})\rangle \displaystyle=\Sigma^{\mu_{1}\nu_{1}}_{1\alpha_{1}\beta_{1}}\langle T^{\alpha_% {1}\beta_{1}}(p_{1})T^{\mu_{2}\nu_{2}}(p_{2})T^{\mu_{3}\nu_{3}}(p_{3})\rangle \displaystyle\langle t^{\mu_{1}\nu_{1}}_{loc}(p_{1})t^{\mu_{2}\nu_{2}}_{loc}(p% _{2})T^{\mu_{3}\nu_{3}}(p_{3})\rangle \displaystyle={\Sigma_{1}}^{\mu_{1}\nu_{1}}_{\alpha_{1}\beta_{1}}{\Sigma_{2}}^% {\mu_{2}\nu_{2}}_{\alpha_{2}\beta_{2}}\langle T^{\alpha_{1}\beta_{1}}(p_{1})T^% {\alpha_{2}\beta_{2}}(p_{2})T^{\mu_{3}\nu_{3}}(p_{3})\rangle \displaystyle\langle t^{\mu_{1}\nu_{1}}_{loc}(p_{1})t^{\mu_{2}\nu_{2}}_{loc}(p% _{2})t_{loc}^{\mu_{3}\nu_{3}}(p_{3})\rangle \displaystyle={\Sigma_{1}}^{\mu_{1}\nu_{1}}_{\alpha_{1}\beta_{1}}{\Sigma_{2}}^% {\mu_{2}\nu_{2}}_{\alpha_{2}\beta_{2}}{\Sigma_{3}}^{\mu_{3}\nu_{3}}_{\alpha_{3% }\beta_{3}}\braket{T^{\alpha_{1}\beta_{1}}(p_{1})T^{\alpha_{2}\beta_{2}}(p_{2}% )T^{\alpha_{3}\beta_{3}}(p_{3})}. (5.8)

Using the projectors \Pi one can write the most general form of the transverse-traceless part as

 \braket{t^{\mu_{1}\nu_{1}}(p_{1})\,t^{\mu_{2}\nu_{2}}(p_{2})\,t^{\mu_{3}\nu_{3% }}(p_{3})}=\Pi^{\mu_{1}\nu_{1}}_{\alpha_{1}\beta_{1}}(p_{1})\Pi^{\mu_{2}\nu_{2% }}_{\alpha_{2}\beta_{2}}(p_{2})\Pi^{\mu_{3}\nu_{3}}_{\alpha_{3}\beta_{3}}(p_{3% })\,\,X^{\alpha_{1}\beta_{1}\,\alpha_{2}\beta_{2}\,\alpha_{3}\beta_{3}}, (5.9)

where X is a general tensor of rank six built from the metric and momenta. One can enumerate all possible tensors that can appear in X, and simplify the expansion by observing that whenever a tensor component of X contains at least one of the following tensors

 \delta^{\alpha_{1}\beta_{1}},\ \delta^{\alpha_{2}\beta_{2}},\ \delta^{\alpha_{% 3}\beta_{3}},\ p_{1}^{\alpha_{1}},\ p_{1}^{\beta_{1}},\ p_{2}^{\alpha_{2}},\ p% _{2}^{\beta_{2}},\ p_{3}^{\alpha_{3}},\ p_{3}^{\beta_{3}} (5.10)

it will vanish after contraction with the projectors, if these carry the same momentum dependence of each of the p_{i}’s . The expansion of X is chosen to be symmetric respect to the p_{i}. In this way one has only to consider the tensors

 p_{2}^{\alpha_{1}},\ p_{2}^{\beta_{1}},\ p_{3}^{\alpha_{2}},\ p_{3}^{\beta_{2}% },\ p_{1}^{\alpha_{3}},\ p_{1}^{\beta_{3}},\ \delta^{\alpha_{2}\alpha_{3}},\ % \delta^{\alpha_{1}\alpha_{2}},\ \delta^{\alpha_{1}\alpha_{3}},\dots (5.11)

and the similar ones with the other indices, and write the most general form of the transverse traceless part as

 \displaystyle\braket{t^{\mu_{1}\nu_{1}}(p_{1})t^{\mu_{2}\nu_{2}}(p_{2})t^{\mu_% {3}\nu_{3}}(p_{3})}=\Pi^{\mu_{1}\nu_{1}}_{\alpha_{1}\beta_{1}}(p_{1})\Pi^{\mu_% {2}\nu_{2}}_{\alpha_{2}\beta_{2}}(p_{2})\Pi^{\mu_{3}\nu_{3}}_{\alpha_{3}\beta_% {3}}(p_{3}) \displaystyle      \times\Big{[}A_{1}\,p_{2}^{\alpha_{1}}p_{2}^{\beta_{1}}p_{3% }^{\alpha_{2}}p_{3}^{\beta_{2}}p_{1}^{\alpha_{3}}p_{1}^{\beta_{3}}+A_{2}\,% \delta^{\beta_{1}\beta_{2}}p_{2}^{\alpha_{1}}p_{3}^{\alpha_{2}}p_{1}^{\alpha_{% 3}}p_{1}^{\beta_{3}}+A_{2}\,(p_{1}\leftrightarrow p_{3})\,\delta^{\beta_{2}% \beta_{3}}p_{3}^{\alpha_{2}}p_{1}^{\alpha_{3}}p_{2}^{\alpha_{1}}p_{2}^{\beta_{% 1}} \displaystyle            +A_{2}\,(p_{2}\leftrightarrow p_{3})\,\delta^{\beta_{% 3}\beta_{1}}p_{1}^{\alpha_{3}}p_{2}^{\alpha_{1}}p_{3}^{\alpha_{2}}p_{3}^{\beta% _{2}}+A_{3}\,\delta^{\alpha_{1}\alpha_{2}}\delta^{\beta_{1}\beta_{2}}p_{1}^{% \alpha_{3}}p_{1}^{\beta_{3}}+A_{3}(p_{1}\leftrightarrow p_{3})\,\delta^{\alpha% _{2}\alpha_{3}}\delta^{\beta_{2}\beta_{3}}p_{2}^{\alpha_{1}}p_{2}^{\beta_{1}} \displaystyle                 +A_{3}(p_{2}\leftrightarrow p_{3})\,\delta^{% \alpha_{3}\alpha_{1}}\delta^{\beta_{3}\beta_{1}}p_{3}^{\alpha_{2}}p_{3}^{\beta% _{2}}+A_{4}\,\delta^{\alpha_{1}\alpha_{3}}\delta^{\alpha_{2}\beta_{3}}p_{2}^{% \beta_{1}}p_{3}^{\beta_{2}}+A_{4}(p_{1}\leftrightarrow p_{3})\,\delta^{\alpha_% {2}\alpha_{1}}\delta^{\alpha_{3}\beta_{1}}p_{3}^{\beta_{2}}p_{1}^{\beta_{3}} \displaystyle                              +A_{4}(p_{2}\leftrightarrow p_{3})% \,\delta^{\alpha_{3}\alpha_{2}}\delta^{\alpha_{1}\beta_{2}}p_{1}^{\beta_{3}}p_% {2}^{\beta_{1}}+A_{5}\delta^{\alpha_{1}\beta_{2}}\delta^{\alpha_{2}\beta_{3}}% \delta^{\alpha_{3}\beta_{1}}\Big{]} (5.12)

where we have used the symmetry properties of the projectors, and the coefficients A_{i}\ i=1,\dots,5, are the form factors, scalar functions of the momentum magnitudes p_{i}^{2}.
Using the decompositions (5.3) and (5) the TTT can be re-expressed in terms of longitudinal and transverse traceless operators

 \displaystyle\braket{T^{\mu_{1}\nu_{1}}\,T^{\mu_{2}\nu_{2}}\,T^{\mu_{3}\nu_{3}}} \displaystyle=\braket{t^{\mu_{1}\nu_{1}}\,t^{\mu_{2}\nu_{2}}\,t^{\mu_{3}\nu_{3% }}}+\braket{t_{loc}^{\mu_{1}\nu_{1}}\,t^{\mu_{2}\nu_{2}}\,t^{\mu_{3}\nu_{3}}}+% \braket{t^{\mu_{1}\nu_{1}}\,t_{loc}^{\mu_{2}\nu_{2}}\,t^{\mu_{3}\nu_{3}}}+% \braket{t^{\mu_{1}\nu_{1}}\,t^{\mu_{2}\nu_{2}}\,t_{loc}^{\mu_{3}\nu_{3}}} \displaystyle       +\braket{t^{\mu_{1}\nu_{1}}_{loc}\,t_{loc}^{\mu_{2}\nu_{2}% }\,t^{\mu_{3}\nu_{3}}}+\braket{t_{loc}^{\mu_{1}\nu_{1}}\,t_{loc}^{\mu_{2}\nu_{% 2}}\,t^{\mu_{3}\nu_{3}}}+\braket{t^{\mu_{1}\nu_{1}}\,t_{loc}^{\mu_{2}\nu_{2}}% \,t_{loc}^{\mu_{3}\nu_{3}}}+\braket{t_{loc}^{\mu_{1}\nu_{1}}\,t_{loc}^{\mu_{2}% \nu_{2}}\,t_{loc}^{\mu_{3}\nu_{3}}} (5.13)

or, equivalently, as

 \displaystyle\braket{T^{\mu_{1}\nu_{1}}\,T^{\mu_{2}\nu_{2}}\,T^{\mu_{3}\nu_{3}}} \displaystyle=\braket{t^{\mu_{1}\nu_{1}}\,t^{\mu_{2}\nu_{2}}\,t^{\mu_{3}\nu_{3% }}}+\braket{T^{\mu_{1}\nu_{1}}\,T^{\mu_{2}\nu_{2}}\,t_{loc}^{\mu_{3}\nu_{3}}}+% \braket{T^{\mu_{1}\nu_{1}}\,t_{loc}^{\mu_{2}\nu_{2}}\,T^{\mu_{3}\nu_{3}}} \displaystyle+\braket{t_{loc}^{\mu_{1}\nu_{1}}\,T^{\mu_{2}\nu_{2}}\,T^{\mu_{3}% \nu_{3}}}-\braket{T^{\mu_{1}\nu_{1}}\,t_{loc}^{\mu_{2}\nu_{2}}\,t_{loc}^{\mu_{% 3}\nu_{3}}}-\braket{t_{loc}^{\mu_{1}\nu_{1}}\,t_{loc}^{\mu_{2}\nu_{2}}\,T^{\mu% _{3}\nu_{3}}} \displaystyle-\braket{t_{loc}^{\mu_{1}\nu_{1}}\,T^{\mu_{2}\nu_{2}}\,t_{loc}^{% \mu_{3}\nu_{3}}}+\braket{t_{loc}^{\mu_{1}\nu_{1}}\,t_{loc}^{\mu_{2}\nu_{2}}\,t% _{loc}^{\mu_{3}\nu_{3}}}. (5.14)

All the terms on the right-hand side, apart from the first one, may be computed by means of transverse and trace WI’s. The exact form of the WI’s varies with the exact definition of the operators involved, but all these terms depend uniquely on 2-point functions.
The main goal of the approach, after introducing the A_{i}’s, is to find the solution of the corresponding scalar equations that they satisfy. These are obtained by acting with the special conformal transformations and the projectors on the representation of the TTT as given by (5).
The action of K on the TTT, after the projection on the transverse traceless component gets simplified. Using the explicit expression

 \displaystyle\Sigma^{\alpha\beta}_{\rho\sigma}=2\frac{p_{\sigma}}{p^{2}}p^{(% \alpha}\delta^{\beta)}_{\rho}-\frac{1}{(d-1)}\frac{p_{\rho}p_{\sigma}}{p^{2}}% \delta^{\alpha\beta}-\frac{d-2}{(d-1)}\frac{p^{\alpha}p^{\beta}p_{\rho}p_{% \sigma}}{(p^{2})^{2}}+\frac{1}{d-1}\delta^{\alpha\beta}\delta_{\rho\sigma}-% \frac{1}{d-1}\delta_{\rho\sigma}\frac{p^{\alpha}p^{\beta}}{p^{2}} (5.15)

and the relation

 \Pi^{\mu\nu}_{\alpha\beta}(p)K(p)^{\kappa}_{scalar}p^{\alpha}p^{\beta}=0 (5.16)

one can show that terms containing two {t_{loc}} operators simplify

 \displaystyle{\Pi_{1}}^{\mu_{1}\nu_{1}}_{\alpha_{1}\beta_{1}}{\Pi_{2}}^{\mu_{2% }\nu_{2}}_{\alpha_{2}\beta_{2}}{\Pi_{3}}^{\mu_{3}\nu_{3}}_{\alpha_{3}\beta_{3}% }{K_{1}}^{\kappa}_{scalar}\left({\Sigma_{1}}^{\alpha_{1}\beta_{1}}_{\rho_{1}% \sigma_{1}}{\Sigma_{3}}^{\alpha_{3}\beta_{3}}_{\rho_{3}\sigma_{3}}{\Pi_{2}}^{% \alpha_{2}\beta_{2}}_{\rho_{2}\sigma_{2}}\langle T^{\rho_{1}\sigma_{1}}T^{\rho% _{2}\sigma_{2}}T^{\rho_{3}\sigma_{3}}\rangle\right) \displaystyle={\Pi_{1}}^{\mu_{1}\nu_{1}}_{\alpha_{1}\beta_{1}}{\Pi_{2}}^{\mu_{% 2}\nu_{2}}_{\alpha_{2}\beta_{2}}{\Pi_{3}}^{\mu_{3}\nu_{3}}_{\alpha_{3}\beta_{3% }}{K_{1}}^{\kappa}_{scalar}\left[\left(2\frac{{p_{1}}_{\sigma_{1}}}{p_{1}^{2}}% p_{1}^{(\alpha_{1}}\delta^{\beta_{1})}_{\rho_{1}}\right)\left(2\frac{{p_{3}}_{% \sigma_{3}}}{{p_{3}}^{2}}{p_{3}}^{(\alpha_{3}}\delta^{\beta_{3})}_{\rho_{3}}% \right){\Pi_{2}}^{\alpha_{2}\beta_{2}}_{\rho_{2}\sigma_{2}}\braket{T^{\rho_{1}% \sigma_{1}}T^{\rho_{2}\sigma_{2}}T^{\rho_{3}\sigma_{3}}}\right] (5.17)

The same identities, applied to the spin part give

 \displaystyle{\Pi_{1}}^{\mu_{1}\nu_{1}}_{\alpha_{1}\beta_{1}}{\Pi_{2}}^{\mu_{2% }\nu_{2}}_{\alpha_{2}\beta_{2}}{\Pi_{3}}^{\mu_{3}\nu_{3}}_{\alpha_{3}\beta_{3}% }{K_{1}}^{\kappa}_{spin}\left({\Sigma_{1}}^{\alpha_{1}\beta_{1}}_{\rho_{1}% \sigma_{1}}{\Sigma_{3}}^{\alpha_{3}\beta_{3}}_{\rho_{3}\sigma_{3}}{\Pi_{2}}^{% \alpha_{2}\beta_{2}}_{\rho_{2}\sigma_{2}}\langle T^{\rho_{1}\sigma_{1}}T^{\rho% _{2}\sigma_{2}}T^{\rho_{3}\sigma_{3}}\rangle\right) \displaystyle=4\,{\Pi_{1}}^{\mu_{1}\nu_{1}}_{\alpha_{1}\beta_{1}}{\Pi_{2}}^{% \mu_{2}\nu_{2}}_{\alpha_{2}\beta_{2}}{\Pi_{3}}^{\mu_{3}\nu_{3}}_{\alpha_{3}% \beta_{3}}\left(\delta^{\kappa}_{\alpha_{1}}\frac{\partial}{\partial p_{1}^{% \lambda}}-\delta^{\kappa\lambda}\frac{\partial}{\partial p_{1}^{\alpha_{1}}}% \right)\left((2\frac{{p_{1}}_{\sigma_{1}}}{p_{1}^{2}}p_{1}^{(\lambda}\delta^{% \beta_{1})}_{\rho_{1}})(2\frac{{p_{3}}_{\sigma_{3}}}{{p_{3}}^{2}}{p_{3}}^{(% \alpha_{3}}\delta^{\beta_{3})}_{\rho_{3}})\times\right. \displaystyle\,\,\,\,\,\,\,\,\,\,\,\,\,\left.\times{\Pi_{3}}^{\alpha_{3}\beta_% {3}}_{\rho_{3}\sigma_{3}}\langle T^{\rho_{1}\sigma_{1}}(p_{1})T^{\rho_{2}% \sigma_{2}}(p_{2})T^{\rho_{3}\sigma_{3}}(p_{3})\rangle\right) (5.18)

where we have explicitly indicated with large round brackets the expression on which the operator acts. As already mentioned, in the equations above the differentiation is performed also on the third momentum p_{3}, which is not independent from p_{1} and p_{2}. Adding (5) and (5) one can show that

 {\Pi_{1}}^{\mu_{1}\nu_{1}}_{\alpha_{1}\beta_{1}}{\Pi_{2}}^{\mu_{2}\nu_{2}}_{% \alpha_{2}\beta_{2}}{\Pi_{3}}^{\mu_{3}\nu_{3}}_{\alpha_{3}\beta_{3}}{K_{1}}^{% \kappa}\langle t_{loc}^{\alpha_{1}\beta_{1}}t^{\alpha_{2}\beta_{2}}t_{loc}^{% \alpha_{3}\beta_{3}}\rangle=0 (5.19)

which can be easily extended to the entire K^{\kappa} operator

 {\Pi_{1}}^{\mu_{1}\nu_{1}}_{\alpha_{1}\beta_{1}}{\Pi_{2}}^{\mu_{2}\nu_{2}}_{% \alpha_{2}\beta_{2}}{\Pi_{3}}^{\mu_{3}\nu_{3}}_{\alpha_{3}\beta_{3}}{K}^{% \kappa}\langle t_{loc}^{\alpha_{1}\beta_{1}}t^{\alpha_{2}\beta_{2}}t_{loc}^{% \alpha_{3}\beta_{3}}\rangle=0 (5.20)

A similar relation can be shown to hold also for the term with three t_{loc} contributions. Proceeding with the reconstruction program, we will be needing the action of the special conformal transformations on correlators with a single t_{loc}, after projection on the transverse traceless sector. In this case the treatment of momenta p_{1} and p_{2} is similar, while in the case of the third (dependent) momentum p_{3} the operatorial action on the tensor structures gets very involved. In order to proceed with the derivation of the scalar equations for the form factors A_{i} we use the Lorentz identity

 \displaystyle{\Pi_{1}}^{\mu_{1}\nu_{1}}_{\alpha_{1}\beta_{1}}{\Pi_{2}}^{\mu_{2% }\nu_{2}}_{\alpha_{2}\beta_{2}}{\Pi_{3}}^{\mu_{3}\nu_{3}}_{\alpha_{3}\beta_{3}% }{K}^{\kappa}_{scalar}\langle t^{\alpha_{1}\beta_{1}}t^{\alpha_{2}\beta_{2}}t_% {loc}^{\alpha_{3}\beta_{3}}\rangle \displaystyle                 ={\Pi_{1}}^{\mu_{1}\nu_{1}}_{\alpha_{1}\beta_{1}% }{\Pi_{2}}^{\mu_{2}\nu_{2}}_{\alpha_{2}\beta_{2}}{\Pi_{3}}^{\mu_{3}\nu_{3}}_{% \alpha_{3}\beta_{3}}4\left[(d-1)\,\delta^{\kappa\alpha_{3}}+{L_{1}}^{\kappa\,% \alpha_{3}}+{L_{2}}^{\kappa\,\alpha_{3}}\right]\left(\frac{{p_{3}}_{\rho_{3}}}% {p_{3}^{2}}\langle t^{\alpha_{1}\beta_{1}}t^{\alpha_{2}\beta_{2}}T^{\rho_{3}% \beta_{3}}\rangle\right) (5.21)

where

and

 \displaystyle{\Pi_{1}}^{\mu_{1}\nu_{1}}_{\alpha_{1}\beta_{1}}{\Pi_{2}}^{\mu_{2% }\nu_{2}}_{\alpha_{2}\beta_{2}}{\Pi_{3}}^{\mu_{3}\nu_{3}}_{\alpha_{3}\beta_{3}% }{K}^{\kappa}_{spin}\langle t^{\alpha_{1}\beta_{1}}t^{\alpha_{2}\beta_{2}}t_{% loc}^{\alpha_{3}\beta_{3}}\rangle \displaystyle  ={\Pi_{1}}^{\mu_{1}\nu_{1}}_{\alpha_{1}\beta_{1}}{\Pi_{2}}^{\mu% _{2}\nu_{2}}_{\alpha_{2}\beta_{2}}{\Pi_{3}}^{\mu_{3}\nu_{3}}_{\alpha_{3}\beta_% {3}}4\left\{\left(\bar{\Sigma}^{\kappa\alpha_{3}}\right)_{\rho_{1}}^{\ \alpha_% {1}}\left[\frac{{p_{3}}_{\rho_{3}}}{p_{3}^{2}}\braket{t^{\rho_{1}\beta_{1}}t^{% \alpha_{2}\beta_{2}}T^{\rho_{3}\beta_{3}}}\right]+\left(\bar{\Sigma}^{\kappa% \alpha_{3}}\right)_{\rho_{2}}^{\ \alpha_{2}}\left[\frac{{p_{3}}_{\rho_{3}}}{p_% {3}^{2}}\braket{t^{\alpha_{1}\beta_{1}}t^{\rho_{2}\beta_{2}}T^{\rho_{3}\beta_{% 3}}}\right]\right\} (5.23)

which allows to derive the projected relations

 \displaystyle\Pi^{\rho_{1}\sigma_{1}}_{\mu_{1}\nu_{1}}\Pi^{\rho_{2}\sigma_{2}}% _{\mu_{2}\nu_{2}}\Pi^{\rho_{3}\sigma_{3}}_{\mu_{3}\nu_{3}}\ {K}^{\kappa}% \braket{t_{loc}^{\mu_{1}\nu_{1}}\,t^{\mu_{2}\nu_{2}}\,t^{\mu_{3}\nu_{3}}} \displaystyle=\Pi^{\rho_{1}\sigma_{1}}_{\mu_{1}\nu_{1}}\Pi^{\rho_{2}\sigma_{2}% }_{\mu_{2}\nu_{2}}\Pi^{\rho_{3}\sigma_{3}}_{\mu_{3}\nu_{3}}\ \left[\mbox{% \small$\displaystyle\frac{4d}{p_{1}^{2}}$}\,\delta^{\kappa\mu_{1}}\,p_{1\alpha% _{1}}\,\braket{\braket{T^{\alpha_{1}\nu_{1}}T^{\mu_{2}\nu_{2}}T^{\mu_{3}\nu_{3% }}}}\right] \displaystyle\Pi^{\rho_{1}\sigma_{1}}_{\mu_{1}\nu_{1}}\Pi^{\rho_{2}\sigma_{2}}% _{\mu_{2}\nu_{2}}\Pi^{\rho_{3}\sigma_{3}}_{\mu_{3}\nu_{3}}\ {K}^{\kappa}% \braket{t^{\mu_{1}\nu_{1}}\,t_{loc}^{\mu_{2}\nu_{2}}\,t^{\mu_{3}\nu_{3}}} \displaystyle=\Pi^{\rho_{1}\sigma_{1}}_{\mu_{1}\nu_{1}}\Pi^{\rho_{2}\sigma_{2}% }_{\mu_{2}\nu_{2}}\Pi^{\rho_{3}\sigma_{3}}_{\mu_{3}\nu_{3}}\ \left[\mbox{% \small$\displaystyle\frac{4d}{p_{2}^{2}}$}\,\delta^{\kappa\mu_{2}}\,p_{2\alpha% _{2}}\,\braket{\braket{T^{\mu_{1}\nu_{1}}T^{\alpha_{2}\nu_{2}}T^{\mu_{3}\nu_{3% }}}}\right] \displaystyle\Pi^{\rho_{1}\sigma_{1}}_{\mu_{1}\nu_{1}}\Pi^{\rho_{2}\sigma_{2}}% _{\mu_{2}\nu_{2}}\Pi^{\rho_{3}\sigma_{3}}_{\mu_{3}\nu_{3}}{K}^{\kappa}\braket{% t^{\mu_{1}\nu_{1}}\,t^{\mu_{2}\nu_{2}}\,t_{loc}^{\mu_{3}\nu_{3}}} \displaystyle=\Pi^{\rho_{1}\sigma_{1}}_{\mu_{1}\nu_{1}}\Pi^{\rho_{2}\sigma_{2}% }_{\mu_{2}\nu_{2}}\Pi^{\rho_{3}\sigma_{3}}_{\mu_{3}\nu_{3}}\ \left[\mbox{% \small$\displaystyle\frac{4d}{p_{3}^{2}}$}\,\delta^{\kappa\mu_{3}}\,p_{3\alpha% _{3}}\,\braket{\braket{T^{\mu_{1}\nu_{1}}T^{\mu_{2}\nu_{2}}T^{\alpha_{3}\nu_{3% }}}}\right] (5.24)

Using this expression, Eq. (4.3) takes the form

 \displaystyle 0 \displaystyle=\Pi^{\rho_{1}\sigma_{1}}_{\mu_{1}\nu_{1}}(p_{1})\Pi^{\rho_{2}% \sigma_{2}}_{\mu_{2}\nu_{2}}(p_{2})\Pi^{\rho_{3}\sigma_{3}}_{\mu_{3}\nu_{3}}(p% _{3})\ \bigg{(}{K}^{\kappa}\,\braket{T^{\mu_{1}\nu_{1}}(p_{1})\,T^{\mu_{2}\nu_% {2}}(p_{2})\,T^{\mu_{3}\nu_{3}}(p_{3})}\bigg{)} \displaystyle=\Pi^{\rho_{1}\sigma_{1}}_{\mu_{1}\nu_{1}}(p_{1})\Pi^{\rho_{2}% \sigma_{2}}_{\mu_{2}\nu_{2}}(p_{2})\Pi^{\rho_{3}\sigma}_{\mu_{3}\nu_{3}}(p_{3})\times \displaystyle\bigg{\{}{K}^{\kappa}\,\braket{t^{\mu_{1}\nu_{1}}(p_{1})\,t^{\mu_% {2}\nu_{2}}(p_{2})\,t^{\mu_{3}\nu_{3}}(p_{3})}+\mbox{\small$\displaystyle\frac% {4d}{p_{1}^{2}}$}\,\delta^{\kappa\mu_{1}}\,p_{1\alpha_{1}}\,\braket{T^{\alpha_% {1}\nu_{1}}(p_{1})T^{\mu_{2}\nu_{2}}(p_{2})T^{\mu_{3}\nu_{3}}(p_{3})} \displaystyle\quad+\mbox{\small$\displaystyle\frac{4d}{p_{2}^{2}}$}\,\delta^{% \kappa\mu_{2}}\,p_{2\alpha_{2}}\,\braket{T^{\mu_{1}\nu_{1}}(p_{1})T^{\alpha_{2% }\nu_{2}}(p_{2})T^{\mu_{3}\nu_{3}}(p_{3})}+\mbox{\small$\displaystyle\frac{4d}% {p_{3}^{2}}$}\,\delta^{\kappa\mu_{3}}\,p_{3\alpha_{3}}\,\braket{T^{\mu_{1}\nu_% {1}}(p_{1})T^{\mu_{2}\nu_{2}}(p_{2})T^{\alpha_{3}\nu_{3}}(p_{3})}\bigg{\}}. (5.25)

The last three terms in the equation above may be re-expressed in terms of 2-point functions via the transverse Ward identities, while the first term in (5.25) can be written as

 \displaystyle\Pi^{\mu_{1}\nu_{1}}_{\alpha_{1}\beta_{1}}(p_{1})\Pi^{\mu_{2}\nu_% {2}}_{\alpha_{2}\beta_{2}}(p_{2})\Pi^{\mu_{3}\nu_{3}}_{\alpha_{3}\beta_{3}}(p_% {3}){K}^{\kappa}\,\braket{t^{\mu_{1}\nu_{1}}(p_{1})\,t^{\mu_{2}\nu_{2}}(p_{2})% \,t^{\mu_{3}\nu_{3}}(p_{3})} \displaystyle=\Pi^{\mu_{1}\nu_{1}}_{\alpha_{1}\beta_{1}}\Pi^{\mu_{2}\nu_{2}}_{% \alpha_{2}\beta_{2}}\Pi^{\mu_{3}\nu_{3}}_{\alpha_{3}\beta_{3}}\times\Bigl{\{}% \,p_{1}^{\kappa}\,\Bigl{(}C_{1,1}\,p_{2}^{\alpha_{1}}p_{2}^{\beta_{1}}p_{3}^{% \alpha_{2}}p_{3}^{\beta_{2}}p_{1}^{\alpha_{3}}p_{1}^{\beta_{3}}+C_{1,2}\,p_{2}% ^{\alpha_{1}}p_{1}^{\alpha_{3}}p_{1}^{\beta_{3}}p_{3}^{\alpha_{2}}\delta^{% \beta_{1}\beta_{2}} \displaystyle\quad+C_{1,3}\,p_{2}^{\beta_{1}}p_{2}^{\alpha_{1}}p_{1}^{\alpha_{% 3}}p_{3}^{\alpha_{2}}\delta^{\beta_{2}\beta_{3}}+C_{1,4}\,p_{2}^{\alpha_{1}}p_% {3}^{\alpha_{2}}p_{1}^{\alpha_{3}}p_{3}^{\beta_{2}}\delta^{\beta_{1}\beta_{3}}% +C_{1,5}\,p_{1}^{\beta_{3}}p_{2}^{\beta_{1}}\delta^{\alpha_{1}\alpha_{2}}% \delta^{\alpha_{3}\beta_{2}} \displaystyle\quad+C_{1,6}\,p_{1}^{\beta_{3}}p_{1}^{\alpha_{3}}\delta^{\alpha_% {1}\alpha_{2}}\delta^{\beta_{1}\beta_{2}}+C_{1,7}\,p_{1}^{\beta_{3}}p_{3}^{% \alpha_{2}}\delta^{\alpha_{3}\alpha_{1}}\delta^{\beta_{1}\beta_{2}}+C_{1,8}\,p% _{2}^{\beta_{1}}p_{2}^{\alpha_{1}}\delta^{\alpha_{2}\alpha_{3}}\delta^{\beta_{% 2}\beta_{3}} \displaystyle\quad+C_{1,9}\,p_{3}^{\beta_{2}}p_{3}^{\alpha_{2}}\delta^{\alpha_% {1}\alpha_{3}}\delta^{\beta_{1}\beta_{3}}+C_{1,10}\,\delta^{\alpha_{1}\beta_{2% }}\delta^{\alpha_{2}\beta_{3}}\delta^{\alpha_{3}\beta_{1}}\Bigr{)}+\big{[}p_{1% }^{\kappa}\leftrightarrow p_{2}^{\kappa};\ C_{1,j}\leftrightarrow C_{2,j}\big{]} \displaystyle\quad\ +\delta^{\kappa\alpha_{1}}\Bigl{(}C_{3,1}\,p_{3}^{\alpha_{% 2}}p_{3}^{\beta_{2}}p_{1}^{\alpha_{3}}p_{1}^{\beta_{3}}p_{2}^{\beta_{1}}+C_{3,% 2}\,\delta^{\alpha_{2}\beta_{3}}p_{3}^{\beta_{2}}p_{1}^{\alpha_{3}}p_{2}^{% \beta_{1}}+C_{3,3}\,\delta^{\alpha_{2}\beta_{1}}p_{1}^{\alpha_{3}}p_{3}^{\beta% _{2}}p_{1}^{\beta_{3}}+C_{3,4}\,\delta^{\alpha_{3}\beta_{1}}p_{3}^{\alpha_{2}}% p_{3}^{\beta_{2}}p_{1}^{\beta_{3}} \displaystyle\qquad\qquad+C_{3,5}\,\delta^{\alpha_{2}\beta_{1}}\delta^{\alpha_% {3}\beta_{2}}p_{1}^{\beta_{3}}+C_{3,6}\,\delta^{\alpha_{2}\beta_{3}}\delta^{% \alpha_{3}\beta_{1}}p_{1}^{\beta_{2}}+C_{3,7}\,\delta^{\alpha_{2}\beta_{3}}% \delta^{\alpha_{3}\beta_{2}}p_{2}^{\beta_{1}}\Bigr{)} \displaystyle\qquad\qquad+[(\alpha_{1},\beta_{1},p_{2})\leftrightarrow(\alpha_% {2},\beta_{2},p_{3});\ C_{3,j}\leftrightarrow C_{4,j}]+[(\alpha_{1},\beta_{1},% p_{2})\leftrightarrow(\alpha_{3},\beta_{3},p_{1});\ C_{3,j}\leftrightarrow C_{% 5,j}]\Bigr{\}} (5.26)

where now the C_{ij}’s are differential equations involving the form factors A_{1},\ A_{2},\ A_{3},\ A_{4},\ A_{5} of the \braket{ttt} in (5.12). The equations to solve are obtained by inserting (5.26) into (5.25) and using (3.8).
For any 3-point function, the resulting equations can be divided into two groups, the primary and the secondary CWI’s. The primary are second-order differential equations and appear as the coefficients of transverse or transverse-traceless tensor containing p_{1}^{\kappa}, p_{2}^{\kappa} and p_{3}^{\kappa}, where \kappa is the special index related to the conformal operator {K}^{\kappa}. The remaining equations, following from all other transverse or transverse-traceless terms, are then secondary conformal Ward identities and correspond to first-order differential equations. Notice that the action of K^{\kappa} on the \braket{ttt} is endomorphic on the transverse traceless sector (see [7] for a derivation).
Obviously, one could define equivalent sets of secondary Ward identities by working directly with (3.8), the conservation WI’s for the stress-energy tensor, but this turns out not to be necessary.
For this purpose notice that Eq. (5.25) is the projection of the special CWI into one specific sector, the transverse traceless part. The remaining sectors are associated with at least one \Sigma projector given in (5.2). Its action on K^{\kappa} can be assimilated to the action of the momentum operator P^{\mu}, being the correlator traceless (\Sigma\sim P) in d dimensions. Using the commutation relation

 [{K}^{\kappa},P^{\nu}]=2i(\delta^{\kappa\nu}D+M^{\kappa\nu}), (5.27)

if the correlator satisfies both the dilatation and the Lorentz WI’s then the action of a \Sigma simplifies as P^{\mu}K^{\kappa}=K^{\kappa}P^{\mu} on the traceless solution. This implies that the remaining sectors in the action of K^{\kappa} are equivalent to conservation WI’s. These are already taken into account by the action of the \Pi projectors on K^{\kappa}, as clear from the right hand side of (5.24), which shows that the action of \Pi K^{\kappa} on the local components of the TTT is still local. Combined this information with the fact that K^{\kappa} maps the transverse-traceless sector into itself, by solving tensorially Eq. (5.25) we account for all the conformal constraints, except for the dilatation WI’s which needs to be considered separately. Notice also that the Lorentz and the permutational symmetries are satisfied by construction, while the dilatation WI’s will be diagonal respect to each of the A_{i}’s, as we are going to shown below.

5.1 The dilatation WI

We illustrate the procedure of deriving scalar equations for the form factors in the simpler case of the dilatation WI’s obtained by the decomposition (4.10). In this case it is sufficient to use the decomposition

 \displaystyle\Pi^{\mu_{1}\nu_{1}}_{1\,\alpha_{1}\beta_{1}}\Pi^{\mu_{2}\nu_{2}}% _{2\,\alpha_{2}\beta_{2}}\Pi^{\mu_{3}\nu_{3}}_{3\,\alpha_{3}\beta_{3}}\left(% \sum_{j=1}^{3}\,\Delta_{j}-2d-\sum_{j=1}^{2}\,p_{j}^{\alpha}\mbox{\small$% \displaystyle\frac{\partial}{\partial p_{j}^{\alpha}}$}\right)\braket{T^{% \alpha_{1}\beta_{1}}T^{\alpha_{2}\beta_{2}}T^{\alpha_{3}\beta_{3}}} \displaystyle=\Pi^{\mu_{1}\nu_{1}}_{1\,\alpha_{1}\beta_{1}}\Pi^{\mu_{2}\nu_{2}% }_{2\,\alpha_{2}\beta_{2}}\Pi^{\mu_{3}\nu_{3}}_{3\,\alpha_{3}\beta_{3}}\left(% \sum_{j=1}^{3}\,\Delta_{j}-2d-\sum_{j=1}^{2}\,p_{j}^{\alpha}\mbox{\small$% \displaystyle\frac{\partial}{\partial p_{j}^{\alpha}}$}\right)\bigg{[}\braket{% t^{\alpha_{1}\beta_{1}}t^{\alpha_{2}\beta_{2}}t^{\alpha_{3}\beta_{3}}}+\dots+% \braket{t_{loc}^{\alpha_{1}\beta_{1}}t_{loc}^{\alpha_{2}\beta_{2}}t^{\alpha_{3% }\beta_{3}}_{loc}}\bigg{]}

taking into account the relations

 \begin{split}&\displaystyle\sum_{j=1}^{2}p_{j}^{\alpha}\mbox{\small$% \displaystyle\frac{\partial}{\partial p_{j}^{\alpha}}$}\,\left(\Pi^{\mu_{i}\nu% _{i}}_{\alpha_{i}\beta_{i}}(p_{i})\right)=0,\qquad i=1,2,3\\ &\displaystyle\sum_{j=1}^{2}p_{j}^{\alpha}\mbox{\small$\displaystyle\frac{% \partial}{\partial p_{j}^{\alpha}}$}\,\left(\Sigma^{\mu_{i}\nu_{i}}_{\alpha_{i% }\beta_{i}}(p_{i})\right)=0,\qquad i=1,2,3\end{split} (5.28)

and the orthogonality between \Pi and \Sigma to obtain the relation

 \displaystyle 0 \displaystyle=\Pi^{\mu_{1}\nu_{1}}_{\alpha_{1}\beta_{1}}(p_{1})\Pi^{\mu_{2}\nu% _{2}}_{\alpha_{2}\beta_{2}}(p_{2})\Pi^{\mu_{3}\nu_{3}}_{\alpha_{3}\beta_{3}}(p% _{3})\left(\sum_{j=1}^{3}\,\Delta_{j}-2d-\sum_{j=1}^{2}\,p_{j}^{\alpha}\mbox{% \small$\displaystyle\frac{\partial}{\partial p_{j}^{\alpha}}$}\right)\Bigg{[}A% _{1}\,p_{2}^{\alpha_{1}}p_{2}^{\beta_{1}}p_{3}^{\alpha_{2}}p_{3}^{\beta_{2}}p_% {1}^{\alpha_{3}}p_{1}^{\beta_{3}} \displaystyle\quad+A_{2}\,\delta^{\beta_{1}\beta_{2}}p_{2}^{\alpha_{1}}p_{3}^{% \alpha_{2}}p_{1}^{\alpha_{3}}p_{1}^{\beta_{3}}+A_{2}\,(p_{1}\leftrightarrow p_% {3})\,\delta^{\beta_{2}\beta_{3}}p_{3}^{\alpha_{2}}p_{1}^{\alpha_{3}}p_{2}^{% \alpha_{1}}p_{2}^{\beta_{1}}+A_{2}\,(p_{2}\leftrightarrow p_{3})\,\delta^{% \beta_{3}\beta_{1}}p_{1}^{\alpha_{3}}p_{2}^{\alpha_{1}}p_{3}^{\alpha_{2}}p_{3}% ^{\beta_{2}} \displaystyle\quad+A_{3}(p_{1}\leftrightarrow p_{3})\,\delta^{\alpha_{2}\alpha% _{3}}\delta^{\beta_{2}\beta_{3}}p_{2}^{\alpha_{1}}p_{2}^{\beta_{1}}+A_{3}(p_{2% }\leftrightarrow p_{3})\,\delta^{\alpha_{3}\alpha_{1}}\delta^{\beta_{3}\beta_{% 1}}p_{3}^{\alpha_{2}}p_{3}^{\beta_{2}}+A_{4}\,\delta^{\alpha_{1}\alpha_{3}}% \delta^{\alpha_{2}\beta_{3}}p_{2}^{\beta_{1}}p_{3}^{\beta_{2}} \displaystyle\quad+A_{4}(p_{1}\leftrightarrow p_{3})\,\delta^{\alpha_{2}\alpha% _{1}}\delta^{\alpha_{3}\beta_{1}}p_{3}^{\beta_{2}}p_{1}^{\beta_{3}}+A_{4}(p_{2% }\leftrightarrow p_{3})\,\delta^{\alpha_{3}\alpha_{2}}\delta^{\alpha_{1}\beta_% {2}}p_{1}^{\beta_{3}}p_{2}^{\beta_{1}}+A_{5}\delta^{\alpha_{1}\beta_{2}}\delta% ^{\alpha_{2}\beta_{3}}\delta^{\alpha_{3}\beta_{1}}\Bigg{]} (5.29)

which is equivalent to the equations

 \left[2d+N_{i}-\sum_{j=1}^{3}\Delta_{j}+\sum_{j=1}^{2}\,p_{j}^{\alpha}\mbox{% \small$\displaystyle\frac{\partial}{\partial p_{j}^{\alpha}}$}\right]\,A_{i}(p% _{1},p_{2},p_{3})=0,\quad i=1,2\ldots 5 (5.30)

where N_{i} is the tensorial dimension of A_{i}, i.e. the number of momenta multiplying the form factor A_{i} and the projectors \Pi.

5.2 Primary CWI’s

From the analysis of (5.25) and (5.26), one can find the primary CWIs that are equivalent to the vanishing of the coefficients C_{1j}, C_{2j} and C_{3j} for j=1,\dots,10. In order to write such equations in a simpler form, we need to rearrange the C_{jk} using the dilatation Ward identities. We will illustrate the explicit procedure for the first coefficient C_{11}, being the others similar.
In order to write such equations we perform the change of variables

 \displaystyle\frac{\partial}{\partial p_{1}^{\mu}} \displaystyle=\frac{\partial p_{1}}{\partial p_{1}^{\mu}}\frac{\partial}{% \partial p_{1}}+\frac{\partial p_{2}}{\partial p_{1}^{\mu}}\frac{\partial}{% \partial p_{2}}+\frac{\partial p_{3}}{\partial p_{1}^{\mu}}\frac{\partial}{% \partial p_{3}}=\frac{p_{\,1\mu}}{p_{1}}\frac{\partial}{\partial p_{1}}+\frac{% p_{1\mu}+p_{2\mu}}{p_{3}}\frac{\partial}{\partial p_{3}} (5.31) \displaystyle\frac{\partial}{\partial p_{2}^{\mu}} \displaystyle=\frac{p_{\,2\mu}}{p_{2}}\frac{\partial}{\partial p_{2}}+\frac{p_% {1\mu}+p_{2\mu}}{p_{3}}\frac{\partial}{\partial p_{3}} (5.32)

where p_{i}=\sqrt{p_{i}^{2}},\ i=1,2,3 are momentum magnitudes. The explicit form of the coefficient C_{11} is given by

 C_{11}=-\frac{2}{p_{3}}\left[p_{1}\frac{\partial^{2}}{\partial p_{1}\partial p% _{3}}+p_{2}\frac{\partial^{2}}{\partial p_{2}\partial p_{3}}\right]A_{1}+\frac% {d-1}{p_{1}}\frac{\partial}{\partial p_{1}}A_{1}-\frac{\partial^{2}}{\partial p% _{1}^{2}}A_{1}+\frac{d-9}{p_{3}}\frac{\partial}{\partial p_{3}}A_{1}-\frac{% \partial^{2}}{\partial p_{3}^{2}}A_{1}\,. (5.33)

Differentiating the dilatation Ward identities respect to the momentum magnitude p_{3} we obtain the relation

 D_{n}\frac{\partial}{\partial p_{3}}A_{n}=\left[\frac{\partial p_{1}}{\partial p% _{3}}\frac{\partial}{\partial p_{1}}+p_{1}\frac{\partial^{2}}{\partial p_{1}% \partial p_{3}}+\frac{\partial p_{2}}{\partial p_{3}}\frac{\partial}{\partial p% _{2}}+p_{2}\frac{\partial^{2}}{\partial p_{3}\partial p_{2}}+\frac{\partial p_% {3}}{\partial p_{3}}\frac{\partial}{\partial p_{3}}+p_{3}\frac{\partial^{2}}{% \partial p_{3}\partial p_{3}}\right]A_{n}\,. (5.34)

that can be simplified as

 D_{n}\frac{\partial}{\partial p_{3}}A_{n}=\left[p_{1}\frac{\partial^{2}}{% \partial p_{1}\partial p_{3}}+p_{2}\frac{\partial^{2}}{\partial p_{3}\partial p% _{2}}+p_{3}\frac{\partial^{2}}{\partial p_{3}\partial p_{3}}\right]A_{n}\,, (5.35)

By using (5.35), we can re-expressed the first term in (5.33) as

 -\frac{2}{p_{3}}\left[p_{1}\frac{\partial^{2}}{\partial p_{1}\partial p_{3}}+p% _{2}\frac{\partial^{2}}{\partial p_{3}\partial p_{2}}\right]A_{1}=\frac{\left(% 2-2D_{1}\right)}{p_{3}}\frac{\partial}{\partial p_{3}}A_{1}+2\frac{\partial^{2% }}{\partial p_{3}^{2}}A_{1}\,, (5.36)

recalling that D_{1} is the degree of the corresponding form factor A_{1}, and in this case D_{1}=\Delta_{3}-4. Inserting this result into (5.33) we simplify the form of the differential equation associated to such coefficient

 C_{11}=\left[-\frac{\partial^{2}}{\partial p_{1}^{2}}+\frac{d-1}{p_{1}}\frac{% \partial}{\partial p_{1}}\right]A_{1}+\left[\frac{\partial^{2}}{\partial p_{3}% ^{2}}+\frac{d+1-2\Delta_{3}}{p_{3}}\frac{\partial}{\partial p_{3}}\right]A_{1}\,. (5.37)

At this stage, in order to write the primary CWIs in a simple way, we define the differential operators

 \displaystyle\textup{K}_{i} \displaystyle=\frac{\partial^{2}}{\partial p_{i}^{2}}+\frac{d+1-2\Delta_{i}}{p% _{i}}\frac{\partial}{\partial p_{i}}\qquad i=1,2,3 (5.38a) \displaystyle\textup{K}_{ij} \displaystyle=\textup{K}_{i}-\textup{K}_{j}\,, (5.38b)

where \Delta_{j} is the conformal dimension of the j-th operator in the 3-point function under consideration. Through this definition the C_{11} is re-expressed as

 C_{11}=(\textup{K}_{3}-\textup{K}_{1})A_{1}=\textup{K}_{31}A_{1}\,. (5.39)

The procedure presented above permits us to obtain a simple second-order differential equations and are applied in the same way for all C_{1j}, j=1,2,3.

The primary CWIs are obtained, as previously discussed, when the coefficients C_{1j} and C_{2j} are equal to zero. For instance, for the A_{1} form factor we obtain

Note that, from the definition (5.38), we have

for any i,j,k={1,2,3}. One can therefore subtract corresponding pairs of equations and obtain the following system of independent partial differential equations

Since in the \braket{TTT} \Delta_{1}=\Delta_{2}=\Delta_{3}=d, using the manipulations discussed above one obtains all the primary CWIs for the form factors A_{i} in the form

 \begin{aligned} &\displaystyle K_{13}A_{1}=0\\ &\displaystyle K_{13}A_{2}=8A_{1}\\ &\displaystyle K_{13}A_{2}(p_{1}\leftrightarrow p_{3})=-8A_{1}\\ &\displaystyle K_{13}A_{2}(p_{2}\leftrightarrow p_{3})=0\\ &\displaystyle K_{13}A_{3}=2A_{2}\\ &\displaystyle K_{13}A_{3}(p_{1}\leftrightarrow p_{3})=-2A_{2}(p_{1}% \leftrightarrow p_{3})\\ &\displaystyle K_{13}A_{3}(p_{2}\leftrightarrow p_{3})=0\\ &\displaystyle K_{13}A_{4}=-4A_{2}(p_{2}\leftrightarrow p_{3})\\ &\displaystyle K_{13}A_{4}(p_{1}\leftrightarrow p_{3})=4A_{2}(p_{2}% \leftrightarrow p_{3})\\ &\displaystyle K_{13}A_{4}(p_{2}\leftrightarrow p_{3})=4A_{2}(p_{1}% \leftrightarrow p_{3})-4A_{2}\\ &\displaystyle K_{13}A_{5}=2\left[A_{4}-A_{4}(p_{1}\leftrightarrow p_{3})% \right]\\ \end{aligned}\qquad\begin{aligned} &\displaystyle K_{23}A_{1}=0\\ &\displaystyle K_{23}A_{2}=8A_{1}\\ &\displaystyle K_{23}A_{2}(p_{1}\leftrightarrow p_{3})=0\\ &\displaystyle K_{23}A_{2}(p_{2}\leftrightarrow p_{3})=-8A_{1}\\ &\displaystyle K_{23}A_{3}=2A_{2}\\ &\displaystyle K_{23}A_{3}(p_{1}\leftrightarrow p_{3})=0\\ &\displaystyle K_{23}A_{3}(p_{2}\leftrightarrow p_{3})=-2A_{2}(p_{2}% \leftrightarrow p_{3})\\ &\displaystyle K_{23}A_{4}=-4A_{2}(p_{1}\leftrightarrow p_{3})\\ &\displaystyle K_{23}A_{4}(p_{1}\leftrightarrow p_{3})=4A_{2}(p_{2}% \leftrightarrow p_{3})-4A_{2}\\ &\displaystyle K_{23}A_{4}(p_{2}\leftrightarrow p_{3})=4A_{2}(p_{1}% \leftrightarrow p_{3})\\ &\displaystyle K_{23}A_{5}=2\left[A_{4}-A_{4}(p_{2}\leftrightarrow p_{3})% \right]\\ \end{aligned} (5.43)

As already mentioned above, the solutions of these equations can be obtained by mapping them into an hypergeometric system of equations for the Appell function F_{4}. As shown in [19] each equation is equivalent to a system of two coupled equations with specific indices that we have shown in [7] to be universal. Differently from the case considered in [19] here the system of equations is far more complicated and it has been discussed in [18] in terms of 3K integrals, which are integrals of 3 Bessel functions. As we are going to see, the goal of the next section is to show how it is possible to use a direct method based on the operatorial splitting of the hypergeometric differential operators in order to relate inhomogeneous solutions to the homogeneous ones. This is obtained by re-expressing the operators K_{ij} in terms of other operators \bar{K}_{ij}, which characterize some homogeneous equations, plus extra operators which are first order in the derivative respect to to the momenta. The action of the extra operators on each F_{4} can be rearranged by suitable shifts of the parameters in F_{4} and using the few known properties of this Appell function. The method follows the simpler case discussed in [7], that we extend. We illustrate the approach, leaving to Appendix B the more technical details. One of the difficulties of the system of equations (5.43) is the presence of exchanged momenta on their right hand side which couple all the constants appearing in the solution in a nontrivial way. The equations for each form factor A_{i} define a coupled system of two equations, which are inhomogeneous, except for A_{1}. The inhomogeneous equations are solved, for each form factor A_{2},...A_{5} separately, as a superposition of a particular solution of the inhomogeneous equations and the general solution of the homogeneous ones, with the free independent constant identified at the end of the entire procedure. We have recollected below the main points, leaving some of the more technical details to an appendix.

6 Solutions of the primary CWI’s by an operatorial method

The transition to the system of hypergeometric differential equations which characterize the form factors can be obtained in various ways. For scalar correlators this has been discussed in [19] using in K^{\kappa} the change of variables

 \displaystyle\frac{\partial}{\partial p_{1}^{\mu}} \displaystyle= \displaystyle 2(p_{1\,\mu}+p_{2\,\mu})\frac{\partial}{\partial p_{3}^{2}}+% \frac{2}{p_{3}^{2}}\left((1-x)p_{1\,\mu}-x\,p_{2\,\mu}\right)\frac{\partial}{% \partial x}-2(p_{1\,\mu}+p_{2\,\mu})\frac{y}{p_{3}^{2}}\frac{\partial}{% \partial y}\,, (6.1) \displaystyle\frac{\partial}{\partial p_{2}^{\mu}} \displaystyle= \displaystyle 2(p_{1\,\mu}+p_{2\,\mu})\frac{\partial}{\partial p_{3}^{2}}-2(p_% {1\,\mu}+p_{2\,\mu})\frac{x}{p_{3}^{2}}\frac{\partial}{\partial x}+\frac{2}{p_% {3}^{2}}\left((1-y)p_{2\,\mu}-y\,p_{1\,\mu}\right)\frac{\partial}{\partial y}. (6.2)

with

Here we are taking p_{3} as "pivot" in the expansion, but we could equivalently choose any of the 3 momentum invariants. The hypergeometric character of the CWI’s was recognized independently in [19] and in [18]. Here we are going to briefly overview the derivation of such equations in the case of the TTT, before discussing a direct method of solutions that we have developed for the TJJ in [7] and that we are going to generalize.
As already mentioned, the method exploits the universality of the Fuchsian points of such equations, a property which holds for all the 3-point functions. It is a general characteristics of the CWI’s associated to such correlators, as we have verified in several cases. The solutions of such equations take a form given by the product of the Appell function F_{4} times x and y as given in (6.3), raised at specific powers a,b (indices), which are universal. An overall extra factor of the momentum (p_{3}) raised to a specific power is introduced in such a way to give the correct scaling behaviour of the solution for each form factor A_{i}.
For each system (i.e. each form factor) we first solve the homogeneous equation, determining the general solution. We then add to this a particular solution of the inhomogeneous equation. The latter is obtained by a split of the differential operator K_{ij}, which can be performed in various ways. The split that we adopt in this case is different form the one used in [7].
We start by reviewing briefly the case of the scalar correlator \Phi(p_{1},p_{2},p_{3}) in order to make our discussion self-contained and define our conventions. In this case the equations are homogeneous, of the form

and need to be combined with the scaling equation

 \sum_{i=1}^{3}p_{i}\frac{\partial}{\partial p_{i}}\Phi=(\Delta-2d)\Phi. (6.5)

The ansatz is generated by a combination of a single power of the momentum "pivot" p_{3} and powers of x and y

 \Phi(p_{1},p_{2},p_{3})=p_{3}^{\Delta-2d}x^{a}y^{b}F(x,y). (6.6)

\Phi is required to be homogenous of degree \Delta-2d under a scale transformation, according to (6.5), and in (6.6) this is taken into account by the factor p_{3}^{\Delta-2d}. Inserting the ansatz one derives the equation

 \displaystyle K_{13}\Phi \displaystyle=4(p_{3}^{2})^{\Delta/2-d-1}x^{a}y^{b}\left(x(1-x)\frac{\partial}% {\partial x\partial x}+(Ax+\gamma)\frac{\partial}{\partial x}-2xy\frac{% \partial^{2}}{\partial x\partial y}-y^{2}\frac{\partial^{2}}{\partial y% \partial y}+Dy\frac{\partial}{\partial y}+\left(E+\frac{G}{x}\right)\right) \displaystyle                          \times F(x,y)=0 (6.7)

with

 \displaystyle A=D=\Delta_{1}+\Delta_{2}-1-2a-2b-\frac{3d}{2}\qquad\gamma(a)=2a% +\frac{d}{2}-\Delta_{1}+1 \displaystyle G=\frac{a}{2}(d+2a-2\Delta_{1}) \displaystyle E=-\frac{1}{4}(2a+2b+2d-\Delta_{1}-\Delta_{2}-\Delta_{3})(2a+2b+% d-\Delta_{1}-\Delta_{2}+\Delta_{3}). (6.8)

and

 \displaystyle K_{23}\Phi \displaystyle=4p_{3}^{\Delta-2d-2}x^{a}y^{b}\left(y(1-y)\frac{\partial}{% \partial y\partial y}+(A^{\prime}y+\gamma^{\prime})\frac{\partial}{\partial y}% -2xy\frac{\partial^{2}}{\partial x\partial y}-x^{2}\frac{\partial^{2}}{% \partial x\partial x}+D^{\prime}x\frac{\partial}{\partial x}+\left(E^{\prime}+% \frac{G^{\prime}}{y}\right)\right) \displaystyle                          \times F(x,y)=0 (6.9)

with

Notice that in both equations we need to set G/x=0 and G^{\prime}/y=0 in order to reproduce an hypergeometric system, which sets conditions on the Fuchsian exponents a and b. These are

and

As we have verified, the four independent solutions of the CWI’s are all characterised by the same 4 pairs of indices (a_{i},b_{j}) (i,j=1,2). Our conventions for the parametric dependences in F_{4} are the same of those introduced in [7]

We also have

The general solution for the scalar correlator, for instance, takes the form

 \Phi(p_{1},p_{2},p_{3})=p_{3}^{\Delta-2d-2}\sum_{a,b}c(a,b,\vec{\Delta})\,x^{a% }y^{b}\,F_{4}(\alpha(a,b),\beta(a,b);\gamma(a),\gamma^{\prime}(b);x,y) (6.15)

where the sum runs over the four values a_{i},b_{i} i=0,1 with constants c(a,b,\vec{\Delta}), with \vec{\Delta}=(\Delta_{1},\Delta_{2},\Delta_{3}). We will also define

 \displaystyle\alpha(a_{0},b_{0})=\frac{d}{2}-\frac{\Delta_{1}+\Delta_{2}-% \Delta_{3}}{2}, \displaystyle\beta(b_{0})=d-\frac{\Delta_{1}+\Delta_{2}+\Delta_{3}}{2}, \displaystyle\gamma(a_{0})=\frac{d}{2}+1-\Delta_{1}, \displaystyle\gamma(b_{0})=\frac{d}{2}+1-\Delta_{2}, (6.16)

and the 4 independent solutions can be re-expressed in terms of the parameters above. The solution for a=0 and b=0 is known as Appell’s function F_{4}, a generalized hypergeometric function of two variables [31]

 \displaystyle F_{4}(\alpha(a,b),\beta(a,b);\gamma(a),\gamma^{\prime}(b);x,y)=% \sum_{i=0}^{\infty}\sum_{j=0}^{\infty}\frac{(\alpha(a,b))_{i+j}\,(\beta(a,b))_% {i+j}}{(\gamma(a))_{i}\,(\gamma^{\prime}(b))_{j}}\frac{x^{i}}{i!}\frac{y^{j}}{% j!} (6.17)

where we have used the standard notations (\alpha)_{i}=\Gamma(\alpha+i)/\Gamma(\alpha) for the Pochammer symbol. \alpha\ldots\gamma^{\prime} are the first, second\ldots, fourth parameters of F_{4}. The 4 independent solutions of (6.7) and (6.9) are then all of the form x^{a}y^{b}F_{4}, where the hypergeometric functions will take some specific values for its parameters \alpha(a,b),\beta(a,b)\ldots etc, with a and b fixed by (6.11) and (6.12).
Next, we are going to extend the analysis presented for the TJJ to the TTT using an alternative splitting of the hyergeometric operators K_{ij} in order to deal with the more difficult structure of the global system of equations which should be satisfied by the form factors.

6.1 Form factors: the solution for A_{1}

We start from A_{1} by solving the two equations from (5.43)

In this case we introduce the ansatz

 A_{1}=p_{3}^{\Delta-2d-6}x^{a}y^{b}F(x,y) (6.19)

and derive two hypergeometric equations as previously, which are characterised by the same indices (a_{i},b_{j}) as before in (6.11) and (6.12), but new values of the 4 defining parameters. We obtain

 A_{1}(p_{1},p_{2},p_{3})=p_{3}^{\Delta-2d-6}\sum_{a,b}c^{(1)}(a,b,\vec{\Delta}% )\,x^{a}y^{b}\,F_{4}(\alpha(a,b)+3,\beta(a,b)+3;\gamma(a),\gamma^{\prime}(b);x% ,y) (6.20)

with the expression of \alpha(a,b),\beta(a,b),\gamma(a),\gamma^{\prime}(b) as given before

 \displaystyle\alpha(a,b) \displaystyle=a+b+\frac{d}{2}-\frac{1}{2}(\Delta_{2}-\Delta_{3}+\Delta_{1}) \displaystyle\beta(a,b) \displaystyle=a+b+d-\frac{1}{2}(\Delta_{1}+\Delta_{2}+\Delta_{3}) (6.21)

and

 \displaystyle\gamma(a) \displaystyle=2a+\frac{d}{2}-\Delta_{1}+1 \displaystyle\gamma^{\prime}(b) \displaystyle=2b+\frac{d}{2}-\Delta_{2}+1. (6.22)

Expressing the values of the scaling dimensions \Delta_{1}=\Delta_{2}=\Delta_{3}=d, then

 \displaystyle A_{1}(p_{1},p_{2},p_{3})=p_{3}^{d-6}\sum_{a,b}c^{(1)}(a,b)\,x^{a% }y^{b}\,F_{4}(\alpha(a,b)+3,\beta(a,b)+3;\gamma(a),\gamma^{\prime}(b);x,y) (6.23)

where now

 \displaystyle a=0,\frac{d}{2}, \displaystyle b=0,\frac{d}{2}, \displaystyle\alpha(a,b)=a+b, \displaystyle\beta(a,b)=a+b-\frac{d}{2}, \displaystyle\gamma(a)=2a-\frac{d}{2}+1, \displaystyle\gamma^{\prime}(b)=2b-\frac{d}{2}+1. (6.24)

One can implement the symmetry condition on the A_{1} form factor which has to be completely symmetric in the exchange of (p_{1},p_{2},p_{3}). The three conditions

 \displaystyle A_{1}(p_{1},p_{3},p_{2}) \displaystyle=A_{1}(p_{1},p_{2},p_{3}) \displaystyle A_{1}(p_{3},p_{2},p_{1}) \displaystyle=A_{1}(p_{1},p_{2},p_{3}) \displaystyle A_{1}(p_{2},p_{1},p_{3}) \displaystyle=A_{1}(p_{1},p_{2},p_{3}) (6.25)

constrain the coefficient c^{(1)}(a,b) and in particular we obtain

 \displaystyle c^{(1)}\left(\frac{d}{2},0\right) \displaystyle=c^{(1)}\left(0,\frac{d}{2}\right) \displaystyle c^{(1)}(0,0) \displaystyle=-\frac{(d-4)(d-2)}{(d+2)(d+4)}c^{(1)}\left(0,\frac{d}{2}\right) \displaystyle c^{(1)}\left(\frac{d}{2},\frac{d}{2}\right) \displaystyle=\frac{\Gamma\left(-\frac{d}{2}\right)\Gamma\left(d+3\right)}{2\,% \Gamma\left(\frac{d}{2}\right)}c^{(1)}\left(0,\frac{d}{2}\right) (6.26)

generating a solution which depends only on one arbitrary constant that we identify as C_{1}

 c^{(1)}\left(0,\frac{d}{2}\right)=C_{1}. (6.27)

6.2 The solution for A_{2} and the operatorial shifts

The equation for A_{2} is inhomogeneous, but the solution can be identified using some properties of the hypergeometric forms of such equations. We recall that in this case they are

 \displaystyle K_{13}A_{2} \displaystyle=8A_{1} (6.28) \displaystyle\ K_{23}A_{2} \displaystyle=8A_{1}. (6.29)

The ansatz which is in agreement with the scaling behaviour of A_{2} in this case is

 A_{2}(p_{1},p_{2},p_{3})=p_{3}^{d-4}\,x^{a}\,y^{b}\,F(x,y). (6.30)

At this stage we proceed with the splitting. We observe that the action of K_{13} and K_{23} on A_{2} can be rearranged as follows

 \displaystyle K_{13}A_{2} \displaystyle=4x^{a}y^{b}p_{3}^{d-6}\bigg{(}\bar{K}_{13}F(x,y)+x\frac{\partial% }{\partial x}F(x,y)+y\frac{\partial}{\partial y}F(x,y)+\bar{\beta}F(x,y)\bigg{)} (6.31) \displaystyle K_{23}A_{2} \displaystyle=4x^{a}y^{b}p_{3}^{d-6}\bigg{(}\bar{K}_{23}F(x,y)+x\frac{\partial% }{\partial x}F(x,y)+y\frac{\partial}{\partial y}F(x,y)+\bar{\beta}F(x,y)\bigg{)} (6.32)

where

 \displaystyle\bar{K}_{13}F(x,y) \displaystyle=\bigg{\{}x(1-x)\frac{\partial^{2}}{\partial x^{2}}-y^{2}\frac{% \partial^{2}}{\partial y^{2}}-2\,x\,y\frac{\partial^{2}}{\partial x\partial y}% +\big{[}\gamma(a)-(\tilde{\alpha}(a,b)+\bar{\beta}(a,b)+1)x\big{]}\frac{% \partial}{\partial x} \displaystyle                                           +\frac{a(a-a_{1})}{x}-% (\tilde{\alpha}(a,b)+\bar{\beta}(a,b)+1)y\frac{\partial}{\partial y}-\tilde{% \alpha}(a,b)\,\bar{\beta}(a,b)\bigg{\}}F(x,y) (6.33)

and

 \displaystyle\bar{K}_{23}A_{2} \displaystyle=\bigg{\{}y(1-y)\frac{\partial^{2}}{\partial y^{2}}-x^{2}\frac{% \partial^{2}}{\partial x^{2}}-2\,x\,y\frac{\partial^{2}}{\partial x\partial y}% +\big{[}\gamma^{\prime}(b)-(\tilde{\alpha}(a,b)+\bar{\beta}(a,b)+1)y\big{]}% \frac{\partial}{\partial y} \displaystyle                                           +\frac{b(b-b_{1})}{y}-% (\tilde{\alpha}(a,b)+\bar{\beta}(a,b)+1)x\frac{\partial}{\partial x}-\tilde{% \alpha}(a,b)\,\bar{\beta}(a,b)\bigg{\}}F(x,y) (6.34)

with

At this point we notice that the hypergeometric function that satisfy the system of equations

 \left\{\begin{split}\displaystyle\bar{K}_{23}F(x,y)&\displaystyle=0\\ \displaystyle\bar{K}_{13}F(x,y)&\displaystyle=0\\ \end{split}\right. (6.36)

can be taken of the form

 \Phi_{1}^{(2)}(x,y)=\sum_{a,b}c^{(2)}_{1}(a,b)\,x^{a}y^{b}\,F_{4}(\alpha(a,b)+% 3,\beta(a,b)+2;\gamma(a),\gamma^{\prime}(b);x,y) (6.37)

with c^{(2)}_{1} constant depending on the parameters a,b fixed at the ordinary values (a_{0},b_{0}),(a_{1},b_{0}),(a_{0},b_{1}) and (a_{1},b_{1}) as in the previous cases (6.11) and (6.12). The convention that we adopt on the indices appearing on the constants is as follows.
The superscript (i) on the constant c^{(i)}, is the index of the corresponding form factors and it is used in the homogeneous solution of the corresponding set of equations. On the other hand, the subscript j, in the constant c^{(i)}_{j} instead, specifies the particular (inhomogeneous) solution of the same system of equations for the form factor A_{i}.
For instance, if one considers the particular solution \Phi_{1}^{(2)} in (6.37), the constant c_{1}^{(2)} is well defined using this convention. In fact it tells us that this solution is the first particular solution of the inhomogeneous set of equations for the A_{2} form factors. It is worth mentioning that all these constants will be fixed, at the end, just in terms of the homogeneous ones that don’t carry any subscript.
As previously remarked, the values of the exponents a and b remain the same for any equation involving either a K_{i,j} or a \bar{K}_{ij}, as one can verify.
At this point, to show that \Phi_{1}^{(2)} is a solution of Eqs. (6.28) we use the property

 \frac{\partial^{p+q}F_{4}(a,b;c_{1},c_{2};x,y)}{\partial x^{p}\partial y^{q}}=% \frac{(a)_{p+q}(b)_{p+q}}{(c_{1})_{p}(c_{2})_{q}}F_{4}(a+p+q,b+p+q;c_{1}+p;c_{% 2}+q;x,y) (6.38)

using the Pochammer symbol previously defined, from which one derives the simpler relations

 \displaystyle\frac{\partial F_{4}(a,b;c_{1},c_{2};x,y)}{\partial x}=\frac{ab}{% c_{1}}F_{4}(a+1,b+1,c_{1}+1,c_{2},x,y) \displaystyle\frac{\partial F_{4}(a,b;c_{1},c_{2};x,y)}{\partial y}=\frac{ab}{% c_{2}}F_{4}(a+1,b+1,c_{1},c_{2}+1,x,y). (6.39)

We will be also using the known relation on the shift of one parameter of F_{4}

 F_{4}(a,b,c_{1}-1,c_{2};x,y)=F_{4}(a,b,c_{1},c_{2},x,y)+x\frac{\partial}{% \partial\,x}\,F_{4}(a,b,c_{1},c_{2},x,y) (6.40)

 x\,F_{4}(a+1,b+1,c_{1},c_{2};x,y)=\frac{(c_{1}-1)(c_{1}-2)}{a\,b}\bigg{[}F_{4}% (a,b,c_{1}-2,c_{2},x,y)-F_{4}(a,b,c_{1}-1,c_{2},x,y)\bigg{]}, (6.41)

and furthermore the symmetry relation

 \displaystyle F_{4}(\alpha,\beta;\gamma,\gamma^{\prime};x,y) \displaystyle=\frac{\Gamma(\gamma^{\prime})\Gamma(\beta-\alpha)}{\Gamma(\gamma% ^{\prime}-\alpha)\Gamma(\beta)}(-y)^{-\alpha}\,F_{4}\left(\alpha,\alpha-\gamma% ^{\prime}+1;\gamma,\alpha-\beta+1;\frac{x}{y},\frac{1}{y}\right) \displaystyle\qquad+\frac{\Gamma(\gamma^{\prime})\Gamma(\alpha-\beta)}{\Gamma(% \gamma^{\prime}-\beta)\Gamma(\alpha)}(-y)^{-\beta}\,F_{4}\left(\beta-\gamma^{% \prime}+1,\beta;\gamma,\beta-\alpha+1;\frac{x}{y},\frac{1}{y}\right)\,. (6.42)

already used in [19] in the analysis of a scalar case, in order to impose the symmetry under the exchange of two of the three momenta. All these relations can be used in order to consider the action of K_{13} and K_{23} on the the \Phi_{2}^{(1)} in (6.31) and in (6.32), obtaining

 \displaystyle K_{13}\Phi_{1}^{(2)}(x,y)=4p_{3}^{d-6}\sum_{a,b}c^{(2)}_{1}(a,b)% \,x^{a}y^{b}\bigg{(}x\frac{\partial}{\partial x}+y\frac{\partial}{\partial y}+% (\beta+2)\bigg{)}\,F_{4}(\alpha+3,\beta+2;\gamma,\gamma^{\prime};x,y) \displaystyle=4p_{3}^{d-6}\sum_{a,b}c^{(1)}_{2}(a,b)\,x^{a}y^{b}\bigg{[}(% \alpha+3)\bigg{(}x\frac{(\beta+2)}{\gamma}F_{4}(\alpha+4,\beta+3;\gamma+1,% \gamma^{\prime};x,y) \displaystyle         +y\frac{(\beta+2)}{\gamma^{\prime}}F_{4}(\alpha+4,\beta+% 3;\gamma,\gamma^{\prime}+1;x,y)\bigg{)}+(\beta+2)\,F_{4}(\alpha+3,\beta+2;% \gamma,\gamma^{\prime};x,y)\bigg{]} (6.43)

where, for simplicity, we have denoted with \alpha=\alpha(a,b) and \beta=\beta(a,b). At this point, using the following properties of hypergeometric functions [31]

 \displaystyle\frac{b}{c_{1}}\,x\,F_{4}(a+1,b+1,c_{1}+1,c_{2},x,y)+\frac{b}{c_{% 2}}\,y\,F_{4}(a+1,b+1,c_{1},c_{2}+1,x,y) \displaystyle=F_{4}(a+1,b,c_{1},c_{2},x,y) \displaystyle-F_{4}(a,b,c_{1},c_{2},x,y)aF_{4}(a+1,b,c_{1},c_{2},x,y)-bF_{4}(a% ,b+1,c_{1},c_{2},x,y) \displaystyle=(a-b)F_{4}(a,b,c_{1},c_{2},x,y) (6.44)

after some algebra, it is simple to verify that

 \displaystyle K_{13}\Phi_{1}^{(2)}(x,y) \displaystyle=4p_{3}^{d-6}\sum_{a,b}c^{(2)}_{1}(a,b)\,x^{a}y^{b}\big{(}\beta(a% ,b)+2\big{)}F_{4}(\alpha+3,\beta+3,\gamma,\gamma^{\prime},x,y) (6.45)

and in the same way

 \displaystyle K_{23}\Phi_{1}^{(2)}(x,y)=4p_{3}^{d-6}\sum_{a,b}c^{(2)}_{1}(a,b)% \,x^{a}y^{b}\big{(}\beta(a,b)+2\big{)}F_{4}(\alpha+3,\beta+3,\gamma,\gamma^{% \prime},x,y). (6.46)

The non-zero right-hand-side of both equations are proportional to the form factor A_{1} given in (6.20). Once this particular solution is determined, (6.20), by comparison, gives the conditions on c_{1}^{(2)} as

 \displaystyle c_{1}^{(2)}(a,b) \displaystyle=\frac{2}{\big{(}\beta(a,b)+2\big{)}}\,c^{(1)}(a,b). (6.47)

Notice that the coefficient c_{1}^{(2)} of the first particular solution of the inhomogeneous set of equations for A_{2} is fixed in terms of the coefficient c^{(1)} of homogeneous one of A_{1}.

Finally, we obtain the general solution for A_{2} in the TTT case (in which \gamma=\gamma^{\prime} ) by superposing the solution of the homogeneous system generated by (6.28) and (6.29) and the inhomogeneous one (6.37), with a condition on the constants given by (6.47). Therefore, the general expression of the solution for A_{2} is given by

 \displaystyle A_{2} \displaystyle=p_{3}^{d-4}\sum_{ab}x^{a}y^{b}\left[c^{(2)}(a,b)\,F_{4}(\alpha+2% ,\beta+2;\gamma,\gamma^{\prime};x,y)+\frac{2\,c^{(1)}(a,b)}{\big{(}\beta+2\big% {)}}F_{4}(\alpha+3,\beta+2;\gamma,\gamma^{\prime};x,y)\right] (6.48)

where also in this case we have used a short-hand notation \alpha=\alpha(a,b), \beta=\beta(a,b), \gamma=\gamma(a) and \gamma^{\prime}=\gamma^{\prime}(b).
Let’s discuss now the symmetry properties of the A_{2} form factors. The latter in fact is symmetric under the exchange p_{1}\leftrightarrow p_{2}, and this condition has to be implemented in the form

 A_{2}(p_{2},p_{1},p_{3})=A_{2}(p_{1},p_{2},p_{3}). (6.49)

Using the properties of the hypergeometric functions previously written, such symmetry constraint relates c^{(1)} and c^{(2)} for the 4 indices a,b which label the homogeneous solutions in the form

 c^{(2)}\left(\frac{d}{2},0\right)=c^{(2)}\left(0,\frac{d}{2}\right) (6.50)
 c^{(1)}\left(\frac{d}{2},0\right)=c^{(1)}\left(0,\frac{d}{2}\right) (6.51)

Notice that of the two equations above, the second is redundant since it is already present as a symmetry condition on A_{1}, as clear from (6.26). Only the first condition on c^{(2)} is new.
We have already established that A_{1} can be written in terms of only a single constant C_{1}, as evident from (6.26) and (6.27). From the expression of A_{2} in (6.48) and using the property (6.50), we can deduce that so far this form factors can be written in terms of four constants: C_{1},\ c^{(2)}\left(0,d/2\right),\ c^{(2)}\left(d/2,d/2\right),\ c^{(2)}\left% (0,0\right). We will see that the symmetry condition on A_{5} will put additional constraint on the coefficients of A_{2}, by allowing us to write this form factors in terms of only two independent constants. At the end, we will see that this iterative method will allow to identify a rather small set of independent constants for each form factor and the entire solution.

6.3 The solution for A_{3}

Also in this case the system of two equations is inhomogeneous

 \left\{\begin{split}\displaystyle K_{13}A_{3}&\displaystyle=2A_{2}\\ \displaystyle K_{23}A_{3}&\displaystyle=2A_{2}\end{split}\right. (6.52)

Using the same strategy of the previous section, it is possible to find two particular solutions of such system using an operatorial split as above

 \displaystyle\Phi_{1}^{(3)}(x,y) \displaystyle=p_{3}^{d-2}\sum_{ab}c_{1}^{(3)}(a,b)\,x^{a}y^{b}F_{4}(\alpha+2,% \beta+1;\gamma,\gamma^{\prime};x,y) (6.53) \displaystyle\Phi_{2}^{(3)}(x,y) \displaystyle=p_{3}^{d-2}\sum_{ab}c_{2}^{(3)}(a,b)\,x^{a}y^{b}F_{4}(\alpha+3,% \beta+1;\gamma,\gamma^{\prime};x,y) (6.54)

and the action of K_{13} and K_{23} on them are respectively

 \displaystyle\left\{\begin{matrix}K_{13}\Phi^{(3)}_{1}=4p_{3}^{d-4}\sum_{ab}x^% {a}y^{b}\,c_{1}^{(3)}(a,b)\,(\beta+1)\,F_{4}(\alpha+2,\beta+2,\gamma,\gamma^{% \prime},x,y)\\ K_{23}\Phi^{(3)}_{1}=4p_{3}^{d-4}\sum_{ab}x^{a}y^{b}\,c_{1}^{(3)}(a,b)\,(\beta% +1)\,F_{4}(\alpha+2,\beta+2,\gamma,\gamma^{\prime},x,y)\end{matrix}\right. (6.55) \displaystyle\left\{\begin{matrix}K_{13}\Phi^{(3)}_{2}=8p_{3}^{d-4}\sum_{ab}x^% {a}y^{b}\,c_{2}^{(3)}(a,b)\,(\beta+1)\,F_{4}(\alpha+3,\beta+2,\gamma,\gamma^{% \prime},x,y)\\ K_{23}\Phi^{(3)}_{2}=8p_{3}^{d-4}\sum_{ab}x^{a}y^{b}\,c_{2}^{(3)}(a,b)\,(\beta% +1)\,F_{4}(\alpha+3,\beta+2,\gamma,\gamma^{\prime},x,y).\end{matrix}\right. (6.56)

These equations have to be equal to the right hand side of (6.52), and this condition fixes the integration constants to be those appearing in A_{2} as

 \displaystyle c_{1}^{(3)}(a,b) \displaystyle=\frac{1}{2(\beta+1)}\,c^{(2)}(a,b) (6.57) \displaystyle c_{2}^{(3)}(a,b) \displaystyle=\frac{1}{2(\beta+1)(\beta+2)}\,c^{(1)}(a,b). (6.58)

The general solution for A_{3} can be obtained by adding to the particular solution above the general solution of the homogeneous system (6.52), for which

 \displaystyle A_{3} \displaystyle=p_{3}^{d-2}\,\sum_{ab}\,x^{a}\,y^{b}\Big{[}c^{(3)}(a,b)\,F_{4}(% \alpha+1,\beta+1,\gamma,\gamma^{\prime};x,y) \displaystyle             +\frac{1}{2(\beta+1)}\,c^{(2)}(a,b)\,F_{4}(\alpha+2,% \beta+1,\gamma,\gamma^{\prime};x,y) \displaystyle             +\frac{1}{2(\beta+1)(\beta+2)}\,c^{(1)}(a,b)F_{4}(% \alpha+3,\beta+1,\gamma,\gamma^{\prime},x,y)\Big{]}. (6.59)

Imposing the symmetry condition on A_{3} under the change p_{1}\leftrightarrow p_{2}

 A_{3}(p_{2},p_{1},p_{3})=A_{3}(p_{1},p_{2},p_{3}) (6.60)

we obtain additional constraints on the homogeneous coefficients c^{(i)}(a,b), i=1,2,3 as

 \displaystyle c^{(3)}\left(\frac{d}{2},0\right)=c^{(3)}\left(0,\frac{d}{2}\right) (6.61) \displaystyle c^{(2)}\left(\frac{d}{2},0\right)=c^{(2)}\left(0,\frac{d}{2}\right) (6.62) \displaystyle c^{(1)}\left(\frac{d}{2},0\right)=c^{(1)}\left(0,\frac{d}{2}% \right). (6.63)

We observe that the last two conditions are already satisfied by the solutions of A_{1} and A_{2} and the new information follows from the first of the equations in (6.61). At this stage the independent constants appearing in A_{3} are seven, but this number will be reduced to three once that the symmetry condition on A_{5} will be also taken into account.

6.4 The solution for A_{4}

The solutions by our method for the A_{4} and A_{5} form factors require a special treatment, due to exchanged momenta on the functional dependence of the form factors on the right hand side of the respective equations. This complication is not present in the case of the TJJ [7]. In particular, the primary WI’s

 \left\{\begin{split}\displaystyle K_{13}A_{4}&\displaystyle=-4A_{2}(p_{2}% \leftrightarrow p_{3})\\ \displaystyle K_{23}A_{4}&\displaystyle=-4A_{2}(p_{1}\leftrightarrow p_{3})% \end{split}\right. (6.64)

involve the symmetrization A_{2}(p_{2}\leftrightarrow p_{3}), that can be obtained from (6.48) with the exchange (p_{2},\Delta_{2})\leftrightarrow(p_{3},\Delta_{3}) and the replacements

 \begin{split}&\displaystyle x\to\tilde{x}=\mbox{\small$\displaystyle\frac{x}{y% }$},\qquad y\to\tilde{y}=\mbox{\small$\displaystyle\frac{1}{y}$},\\ &\displaystyle\alpha(a,b)\to\tilde{\alpha}(a,b)=a+b+\mbox{\small$\displaystyle% \frac{d}{2}$}-\mbox{\small$\displaystyle\frac{\Delta_{1}-\Delta_{2}+\Delta_{3}% }{2}$}=\alpha(a,b)-(\Delta_{2}-\Delta_{3})\\ &\displaystyle\beta(a,b)\to\tilde{\beta}(a,b)=a+b+d-\mbox{\small$\displaystyle% \frac{\Delta_{1}+\Delta_{2}+\Delta_{3}}{2}$}=\beta(a,b)\\ &\displaystyle\gamma(a)\to\tilde{\gamma}(a)=2a+\mbox{\small$\displaystyle\frac% {d}{2}$}-\Delta_{1}+1=\gamma(a)\\ &\displaystyle\gamma^{\prime}(b)\to\tilde{\gamma}^{\prime}(b)=2b+\mbox{\small$% \displaystyle\frac{d}{2}$}-\Delta_{3}+1=\gamma^{\prime}(b)-(\Delta_{2}-\Delta_% {3})\\ \end{split} (6.65)

in the basic solution F_{4}. In the TTT case, inserting the corresponding scaling dimensions, one has \tilde{\alpha}(a,b)=\alpha(a,b) and \tilde{\gamma}^{\prime}(a,b)=\gamma^{\prime}(a,b).
The two particular solutions of the inhomogeneous equations (6.64) can be expressed in the form

 \begin{split}\displaystyle\Phi_{1}^{(4)}&\displaystyle=p_{3}^{d-2}\sum_{ab}c_{% 1}^{(4)}(a,b)\,x^{a}y^{b}F_{4}(\alpha+2,\beta+1;\gamma,\gamma^{\prime};x,y)\\ \displaystyle\Phi_{2}^{(4)}&\displaystyle=p_{3}^{d-2}\sum_{ab}c_{2}^{(4)}(a,b)% \,x^{a}y^{b}F_{4}(\alpha+1,\beta+1;\gamma-1,\gamma^{\prime}-1;x,y)\end{split} (6.66)

where the action of K_{13},K_{23} on them gives

 \displaystyle\left\{\begin{matrix}K_{13}\Phi_{1}^{(4)}=4p_{3}^{d-4}\sum_{ab}x^% {a}y^{b}\,c_{1}^{(4)}(a,b)\,(\beta+1)\,F_{4}(\alpha+2,\beta+2,\gamma,\gamma^{% \prime},x,y)\\ K_{23}\Phi_{1}^{(4)}=4p_{3}^{d-4}\sum_{ab}x^{a}y^{b}\,c_{1}^{(4)}(a,b)\,(\beta% +1)\,F_{4}(\alpha+2,\beta+2,\gamma,\gamma^{\prime},x,y)\end{matrix}\right. (6.67) \displaystyle\left\{\begin{matrix}K_{13}\Phi_{2}^{(4)}=4p_{3}^{d-4}\sum_{ab}c_% {2}^{(4)}(a,b)\,\frac{(\alpha+1)(\beta+1)}{(\gamma-1)}\,x^{a}y^{b}F_{4}(\alpha% +2,\beta+2;\gamma,\gamma^{\prime}-1;x,y)\\ K_{23}\Phi_{2}^{(4)}=4p_{3}^{d-4}\sum_{ab}c_{2}^{(4)}(a,b)\,\frac{(\alpha+1)(% \beta+1)}{(\gamma^{\prime}-1)}\,x^{a}y^{b}F_{4}(\alpha+2,\beta+2;\gamma-1,% \gamma^{\prime};x,y).\end{matrix}\right. (6.68)

Writing the previous expressions explicitly for the four fundamental indices a=0,\,d/2 and b=0,\,d/2, one can compare the two solutions in order to extract information about the corresponding constants introduced in (6.66). We relate the various constants using the intermediate steps worked out in appendix B, to which we refer for further details.
This allows us to derive a particular solution of the system of equations of (6.64). The general solution to (6.64) is obtained by adding such particular solution to the general homogenous one, and can be written in the form

 \displaystyle A_{4} \displaystyle=p_{3}^{d-2}\sum_{ab}x^{a}y^{b}\Bigg{\{}c^{(4)}(a,b)\,F_{4}(% \alpha+1,\beta+1,\gamma,\gamma^{\prime},x,y) \displaystyle+c^{(4)}_{1}(a,b)F_{4}(\alpha+2,\beta+1,\gamma,\gamma^{\prime},x,% y)+\,c^{(4)}_{2}(a,b)F_{4}(\alpha+1,\beta+1,\gamma-1,\gamma^{\prime}-1,x,y) \displaystyle+c^{(4)}_{3}(a,b)F_{4}(\alpha+1,\beta+1,\gamma-1,\gamma^{\prime},% x,y)+\,c^{(4)}_{4}(a,b)F_{4}(\alpha+1,\beta+1,\gamma,\gamma^{\prime}-1,x,y)% \Bigg{\}} (6.69)

with the constants c^{(4)}_{i},\ i=1,2,3,4 given in terms of c^{(1)}(a,b) and c^{(2)}(a,b) once we enforce the symmetry constraints, and with \alpha=\alpha(a,b), \beta=\beta(a,b), \gamma=\gamma(a) and \gamma^{\prime}=\gamma^{\prime}(b), for simplicity. The form factor A_{4} is symmetric under the exchange p_{1}\leftrightarrow p_{2}

 A_{4}(p_{1},p_{2},p_{3})=A_{4}(p_{2},p_{1},p_{3}) (6.70)

and leads to the conditions (6.26), (6.50), (6.61) and to

 \displaystyle c^{(4)}\left(\frac{d}{2},0\right)=c^{(4)}\left(0,\frac{d}{2}% \right), \displaystyle c_{3}^{(4)}\left(0,0\right)=c_{4}^{(4)}\left(0,0\right) \displaystyle c_{3}^{(4)}\left(\frac{d}{2},\frac{d}{2}\right)=c_{4}^{(4)}\left% (\frac{d}{2},\frac{d}{2}\right) \displaystyle c_{3}^{(4)}\left(\frac{d}{2},0\right)=c_{4}^{(4)}\left(0,\frac{d% }{2}\right) \displaystyle c_{3}^{(4)}\left(0,\frac{d}{2}\right)=c_{4}^{(4)}\left(\frac{d}{% 2},0\right) (6.71)
 \displaystyle c^{(4)}_{3}\left(\frac{d}{2},0\right) \displaystyle=\frac{d^{2}}{(d+2)(d+4)}c^{(1)}\left(0,\frac{d}{2}\right)+\frac{% d}{(d-2)}c^{(2)}(0,0)-\frac{d}{(d+2)}c^{(2)}\left(0,\frac{d}{2}\right)-c_{3}^{% (4)}\left(0,\frac{d}{2}\right). (6.72)

Using the relation given in the appendix, the general solution can be parameterized as

 \displaystyle\begin{aligned} \displaystyle c_{2}^{(4)}\left(0,0\right)&% \displaystyle=-\frac{d^{2}}{(d+2)(d+4)}C_{1}\\ \displaystyle c_{2}^{(4)}\left(0,\frac{d}{2}\right)&\displaystyle=-\frac{d^{2}% }{(d+2)(d+4)}C_{1}\\ \displaystyle c_{2}^{(4)}\left(\frac{d}{2},0\right)&\displaystyle=c_{2}^{(4)}% \left(0,\frac{d}{2}\right)\\ \displaystyle c_{2}^{(4)}\left(\frac{d}{2},\frac{d}{2}\right)&\displaystyle=% \frac{d\sec\left(\frac{\pi\,d}{2}\right)\,\Gamma\left(1-\frac{d}{2}\right)^{2}% }{(d+2)(d+4)\,\Gamma(-d)}\,C_{1}\end{aligned} (6.73)

6.5 The A_{5} solution

Also in the case of A_{5} we have to repeat the approach presented in Section 6.4. In particular the primary Ward identities for A_{5} is given by

 \left\{\begin{split}\displaystyle K_{13}A_{5}&\displaystyle=2[A_{4}-A_{4}(p_{1% }\leftrightarrow p_{3})]\\ \displaystyle K_{23}A_{5}&\displaystyle=2[A_{4}-A_{4}(p_{2}\leftrightarrow p_{% 3})]\end{split}\right. (6.74)

and this system of equations admit seven particular solutions. Combined with the homogeneous solution they give

 \displaystyle A_{5} \displaystyle=p_{3}^{d}\sum_{ab}x^{a}y^{b}\Bigg{\{}\frac{1}{\beta}\bigg{[}+c_{% 1}^{(5)}(a,b)\,F_{4}(\alpha+1,\beta,\gamma-1,\gamma^{\prime}-1,x,y) \displaystyle                 +c^{(5)}_{2}(a,b)F_{4}(\alpha+1,\beta,\gamma,% \gamma^{\prime}-1,x,y)+\,c^{(5)}_{3}(a,b)F_{4}(\alpha+1,\beta,\gamma,\gamma^{% \prime},x,y) \displaystyle+c^{(5)}_{4}(a,b)F_{4}(\alpha+1,\beta,\gamma-1,\gamma^{\prime},x,% y)+c^{(5)}_{5}(a,b)F_{4}(\alpha,\beta,\gamma-1,\gamma^{\prime},x,y)+c^{(5)}_{6% }(a,b)F_{4}(\alpha,\beta,\gamma,\gamma^{\prime}-1,x,y)\bigg{]} \displaystyle+\frac{1}{\alpha\,\beta}\bigg{[}c^{(5)}_{7}(a,b)F_{4}(\alpha,% \beta,\gamma-1,\gamma^{\prime}-1,x,y)+c^{(5)}(a,b)\,F_{4}(\alpha,\beta,\gamma,% \gamma^{\prime},x,y)\bigg{]}\Bigg{\}}. (6.75)

In particular the coefficients c^{(5)}_{i}, i=1,\dots,4 are fixed by the use (6.74), and imposing the symmetry conditions on A_{5}

 \displaystyle A_{5}(p_{3},p_{2},p_{1}) \displaystyle=A_{5}(p_{1},p_{2},p_{3}) \displaystyle A_{5}(p_{2},p_{1},p_{3}) \displaystyle=A_{5}(p_{1},p_{2},p_{3}) \displaystyle A_{5}(p_{1},p_{3},p_{2}) \displaystyle=A_{5}(p_{1},p_{2},p_{3}). (6.76)

We have left to Appendix B more details on the identification of the independent constants which characterize this solution and the analogous solution for A_{4}. There are 5 constants overall for the system of primary WI’s, in agreement with the result presented in [25], which reduce to 3 after imposing the constraints derived by secondary WI’s. Such additional reduction can be performed as discussed in [25]. We have listed such secondary CWI’s in appendix A.

6.6 Summary

In this section we will briefly summarize the final solutions obtained for all the form factors.

We obtain

 A_{1}=p_{3}^{d-6}\sum_{a,b}C_{1}\,f_{1}(a,b)\,x^{a}y^{b}\,F_{4}(\alpha(a,b)+3,% \beta(a,b)+3;\gamma(a),\gamma^{\prime}(b);x,y) (6.77)
 \displaystyle f_{1}\left(0,\frac{d}{2}\right) \displaystyle=f_{1}\left(\frac{d}{2},0\right)=1 \displaystyle f_{1}(0,0) \displaystyle=-\frac{(d-4)(d-2)}{(d+2)(d+4)} \displaystyle f_{1}\left(\frac{d}{2},\frac{d}{2}\right) \displaystyle=\frac{\Gamma\left(-\frac{d}{2}\right)\Gamma\left(d+3\right)}{2\,% \Gamma\left(\frac{d}{2}\right)}

where f_{1}(a,b) takes four values for the four Fuchsian indices. In this case the function f_{1} can be read from the expressions (6.26);

 \displaystyle A_{2} \displaystyle=p_{3}^{d-4}\sum_{ab}x^{a}y^{b}\bigg{[}C_{2}\,f_{2}(a,b)\,F_{4}(% \alpha+2,\beta+2;\gamma,\gamma^{\prime};x,y) \displaystyle                                               +\frac{2\,C_{1}}{% \big{(}\beta+2\big{)}}\,f_{1}(a,b)\,F_{4}(\alpha+3,\beta+2;\gamma,\gamma^{% \prime};x,y)\bigg{]}. (6.79)
 \displaystyle f_{2}\left(0,0\right) \displaystyle=\frac{d-2}{d+2} \displaystyle f_{2}\left(\frac{d}{2},0\right) \displaystyle=f_{2}\left(0,\frac{d}{2}\right)=1 \displaystyle f_{2}\left(\frac{d}{2},\frac{d}{2}\right) \displaystyle=\frac{\Gamma(-d/2)\Gamma(d+2)}{\Gamma(d/2)} (6.80)

In the same way we write the explicit form of A_{3} using the results in Appendix B.2 as

 \displaystyle A_{3} \displaystyle=p_{3}^{d-2}\,\sum_{ab}\,x^{a}\,y^{b}\Big{[}C_{3}\,f_{3}(a,b,d)\,% F_{4}(\alpha+1,\beta+1,\gamma,\gamma^{\prime};x,y) \displaystyle                              +\frac{C_{2}}{2(\beta+1)}\,f_{2}(a,% b,d)\,F_{4}(\alpha+2,\beta+1,\gamma,\gamma^{\prime};x,y) \displaystyle                                              +\frac{C_{1}}{2(% \beta+1)(\beta+2)}\,f_{1}(a,b,d)F_{4}(\alpha+3,\beta+1,\gamma,\gamma^{\prime},% x,y)\Big{]}, (6.81)

where

 \displaystyle f_{3}\left(\frac{d}{2},0\right) \displaystyle=f_{3}\left(0,\frac{d}{2}\right)=1 \displaystyle f_{3}\left(0,0\right) \displaystyle=-1 \displaystyle f_{3}\left(\frac{d}{2},\frac{d}{2}\right) \displaystyle=\frac{\Gamma(-d/2)\Gamma(d+1)}{\Gamma(d/2)} (6.82)
 \displaystyle A_{4} \displaystyle=p_{3}^{d-2}\sum_{ab}x^{a}y^{b}\Bigg{\{}c^{(4)}(a,b)\,F_{4}(% \alpha+1,\beta+1,\gamma,\gamma^{\prime},x,y) \displaystyle+c^{(4)}_{1}(a,b)F_{4}(\alpha+2,\beta+1,\gamma,\gamma^{\prime},x,% y)+\,c^{(4)}_{2}(a,b)F_{4}(\alpha+1,\beta+1,\gamma-1,\gamma^{\prime}-1,x,y) \displaystyle+c^{(4)}_{3}(a,b)F_{4}(\alpha+1,\beta+1,\gamma-1,\gamma^{\prime},% x,y)+\,c^{(4)}_{4}(a,b)F_{4}(\alpha+1,\beta+1,\gamma,\gamma^{\prime}-1,x,y)% \Bigg{\}} (6.83)
 \displaystyle c^{(4)}\left(\frac{d}{2},0\right) \displaystyle=c^{(4)}\left(0,\frac{d}{2}\right)=C_{4} \displaystyle c^{(4)}\left(0,0\right) \displaystyle=-C_{4} \displaystyle c^{(4)}\left(\frac{d}{2},\frac{d}{2}\right) \displaystyle=\frac{\Gamma\left(-\frac{d}{2}\right)\Gamma(d+1)}{\Gamma\left(% \frac{d}{2}\right)}C_{4}-\frac{d^{2}\,\Gamma\left(-\frac{d}{2}\right)\Gamma(d+% 2)}{(d+1)(d+2)(d+4)\Gamma\left(\frac{d}{2}\right)}C_{1} (6.84) \displaystyle c_{1}^{(4)}\left(\frac{d}{2},0\right) \displaystyle=c^{(4)}_{1}\left(0,\frac{d}{2}\right)=-C_{2} \displaystyle c_{1}^{(4)}\left(0,0\right) \displaystyle=\frac{2}{(d+2)}\,C_{2} \displaystyle c_{1}^{(4)}\left(\frac{d}{2},\frac{d}{2}\right) \displaystyle=-\frac{2\,\Gamma\left(-\frac{d}{2}\right)\,\Gamma\left(d+2\right% )}{(d+2)\,\Gamma\left(\frac{d}{2}\right)}C_{2} (6.85) \displaystyle c_{2}^{(4)}\left(0,0\right) \displaystyle=-\frac{d^{2}}{(d+2)(d+4)}C_{1} \displaystyle c_{2}^{(4)}\left(0,\frac{d}{2}\right) \displaystyle=c_{2}^{(4)}\left(0,\frac{d}{2}\right)=-\frac{d^{2}}{(d+2)(d+4)}C% _{1} \displaystyle c_{2}^{(4)}\left(\frac{d}{2},\frac{d}{2}\right) \displaystyle=-\frac{d^{2}\,\Gamma\left(d+2\right)\Gamma\left(-\frac{d}{2}% \right)}{(d+1)(d+2)(d+4)\,\Gamma\left(\frac{d}{2}\right)}\,C_{1} (6.86) \displaystyle c_{3}^{(4)}\left(0,0\right) \displaystyle=c_{4}^{(4)}\left(0,0\right)=0 \displaystyle c_{3}^{(4)}\left(\frac{d}{2},0\right) \displaystyle=c_{4}^{(4)}\left(0,\frac{d}{2}\right)=0 \displaystyle c_{3}^{(4)}\left(0,\frac{d}{2}\right) \displaystyle=c_{4}^{(4)}\left(\frac{d}{2},0\right)=\frac{d^{2}}{(d+2)(d+4)}C_% {1} \displaystyle c_{3}^{(4)}\left(\frac{d}{2},\frac{d}{2}\right) \displaystyle=c_{4}^{(4)}\left(\frac{d}{2},\frac{d}{2}\right)=\frac{d^{2}\,% \Gamma\left(d+2\right)\Gamma\left(-\frac{d}{2}\right)}{(d+1)(d+2)(d+4)\,\Gamma% \left(\frac{d}{2}\right)}\,C_{1} (6.87)

Finally we give the form of the A_{5} form factors as

 \displaystyle A_{5} \displaystyle=p_{3}^{d}\sum_{ab}x^{a}y^{b}\Bigg{\{}\frac{1}{\beta}\bigg{[}+c_{% 1}^{(5)}(a,b)\,F_{4}(\alpha+1,\beta,\gamma-1,\gamma^{\prime}-1,x,y) \displaystyle                 +c^{(5)}_{2}(a,b)F_{4}(\alpha+1,\beta,\gamma,% \gamma^{\prime}-1,x,y)+\,c^{(5)}_{3}(a,b)F_{4}(\alpha+1,\beta,\gamma,\gamma^{% \prime},x,y) \displaystyle+\,c^{(5)}_{4}(a,b)F_{4}(\alpha+1,\beta,\gamma-1,\gamma^{\prime},% x,y)+c^{(5)}_{5}(a,b)F_{4}(\alpha,\beta,\gamma-1,\gamma^{\prime},x,y)+\,c^{(5)% }_{6}(a,b)F_{4}(\alpha,\beta,\gamma,\gamma^{\prime}-1,x,y)\bigg{]} \displaystyle+\frac{1}{\alpha\,\beta}\bigg{[}c^{(5)}_{7}(a,b)F_{4}(\alpha,% \beta,\gamma-1,\gamma^{\prime}-1,x,y)+c^{(5)}(a,b)\,F_{4}(\alpha,\beta,\gamma,% \gamma^{\prime},x,y)\bigg{]}\Bigg{\}}. (6.88)

where the coefficients are summarized in Appendix B.2. The global solution is fixed up to five independent constants.

7 Lagrangian realizations and reconstruction

In this section we turn to the central aspect of our analysis, which will allow us to extend the results of the TJJ correlator presented in [7] to the TTT. We will be using the free field theory realizations of such correlator in order to study the structure of the conformal Ward identities in momentum space and, in particular, the form of the anomalous Ward identities once the conformal symmetry is broken by the anomaly. We will work in DR and adopt the \overline{MS} renormalization scheme. Our analysis hinges on the correspondence between the exact result obtained by solving the CWI’s and the perturbative one.
It is clear that the general solutions presented in the former section, though derived regardless of any perturbative picture, become completely equivalent to the latter if, for a given spacetime dimension, we have a sufficient number of independent sectors in the Lagrangian realization that allow us to reproduce the general one. For instance, in d=3,4 such correspondence is exact, as already mentioned in the introduction, since the number of constants in the solution coincides wth the number of possible independent sectors in the free field theory.
In principle, one could proceed with an analysis of the general solutions - such as those presented in the previous section - as d\to 4, by going through a very involved process of extraction of the singularities from their general expressions in terms of F_{4}.
However, this can be avoided once the general results for the A_{i}’s are matched to the perturbative ones. As already mentioned, this brings in an important simplification of the final result for the TTT, which is expressed in terms of the simple \log present in B_{0} and the scalar 3-point function C_{0}. The latter is of type F_{4} in d=4, but takes a far simpler form compared to the expressions derived in the previous section

 \displaystyle C_{0}(p_{1}^{2},p_{2}^{2},p_{3}^{2})=\frac{1}{p_{3}^{2}}\Phi(x,y), (7.1)

where the function \Phi(x,y) is defined as [32]

 \displaystyle\Phi(x,y) \displaystyle= \displaystyle\frac{1}{\lambda}\biggl{\{}2[Li_{2}(-\rho x)+Li_{2}(-\rho y)]+\ln% \frac{y}{x}\ln\frac{1+\rho y}{1+\rho x}+\ln(\rho x)\ln(\rho y)+\frac{\pi^{2}}{% 3}\biggr{\}}, (7.2)

with

This has the important implication that the study of the specific unitarity cuts in the diagrammatic expansions of the correlator [4, 22] which are held responsible for the emergence of the anomaly, acquire a simple particle interpretation and are not an artifact of perturbation theory. Once the general correspondence between Lagrangian and non-Lagrangian solutions is established, we will concentrate on showing how renormalization is responsible for the emergence of specific anomaly poles in this correlators. We anticipate that the vertex will separate, after renormalization, into a traceless part and in an anomaly part, following the same pattern of the TJJ [6]. From that point on, one can use just the Feynman expansion to perform complete further analysis of this vertex at one loop, with no loss of generality whatsoever.

7.1 Perturbative sectors

In this section we define our conventions used for the perturbative sectors.
The quantum actions for the scalar and fermion field are respectively

 \displaystyle S_{scalar} \displaystyle=\mbox{\small$\displaystyle\frac{1}{2}$}\int\,d^{d}x\,\sqrt{-g}% \left[g^{\mu\nu}\nabla_{\mu}\phi\nabla_{\nu}\phi-\chi\,R\,\phi^{2}\right] (7.5) \displaystyle S_{fermion} \displaystyle=\mbox{\small$\displaystyle\frac{i}{2}$}\int\,d^{d}x\,e\,e^{\mu}_% {a}\left[\bar{\psi}\gamma^{a}(D_{\mu}\psi)-(D_{\mu}\bar{\psi})\gamma^{a}\psi% \right], (7.6)

where \chi=(d-2)/(4d-4) for a conformally coupled scalar in d dimensions, and R is the Ricci scalar. e^{\mu}_{a} is the vielbein and e its determinant, with the covariant derivative D_{\mu} given by

 D_{\mu}=\partial_{\mu}+\Gamma_{\mu}=\partial_{\mu}+\mbox{\small$\displaystyle% \frac{1}{2}$}\Sigma^{ab}\,e^{\sigma}_{a}\nabla_{\mu}\,e_{b\,\sigma}. (7.7)

The \Sigma^{ab} are the generators of the Lorentz group in the spin 1/2 representation. The Latin indices are related to the flat space-time and the Greek indices to the curved space-time. For d=4 there is an additional conformal field theory described in terms of free abelian vector fields with the action

 S_{abelian}=S_{M}+S_{gf}+S_{gh} (7.8)

where the three contributions are the Maxwell action, the gauge fixing contribution and the ghost action

 \displaystyle S_{M} \displaystyle=-\mbox{\small$\displaystyle\frac{1}{4}$}\int d^{4}x\,\sqrt{-g}\,% F^{\mu\nu}F_{\mu\nu}, (7.9) \displaystyle S_{gf} \displaystyle=-\mbox{\small$\displaystyle\frac{1}{\xi}$}\int d^{4}x\,\sqrt{-g}% \,(\nabla_{\mu}A^{\mu})^{2}, (7.10) \displaystyle S_{gh} \displaystyle=\int d^{4}x\,\sqrt{-g}\,\,\partial^{\mu}\bar{c}\,\partial_{\mu}% \,c. (7.11)

The computation of the vertices of each theory can be done by taking (at most) two functional derivatives of the action with respect to the metric, since the vacuum expectation values of the third derivatives correspond to massless tadpoles, which are zero in DR. They are given in Fig. 1 and their explicit expressions have been collected in the Appendix D.

Since we are interested in the most general Lagrangian realization of the \braket{TTT} correlator in the conformal case, this can be obtained only by considering the scalar and fermion sectors in general d dimensions.

7.2 Scalar sector

We start from the scalar sector. In the one-loop approximation the contributions to the correlation function are given by the diagrams in Fig. 2. Using the Feynman rules listed in Appendix D, we calculate all the terms in the defining relation of the TTT Eq. (2.4) in momentum space, for the scalar sector, as

 \displaystyle\braket{T^{\mu_{1}\nu_{1}}(p_{1})T^{\mu_{2}\nu_{2}}(p_{2})T^{\mu_% {3}\nu_{3}}(p_{3})}_{S}=\,-V_{S}^{\mu_{1}\nu_{1}\mu_{2}\nu_{2}\mu_{3}\nu_{3}}(% p_{1},p_{2},p_{3})+\sum_{i=1}^{3}W_{S,i}^{\mu_{1}\nu_{1}\mu_{2}\nu_{2}\mu_{3}% \nu_{3}}(p_{1},p_{2},p_{3}) (7.12)

where V_{S} is related to the triangle diagrams in Fig. 2 and W_{S,i} terms are the three bubble contributions labelled by the index i, with i=1,2,3. These contribution are given by

 \displaystyle V_{S}^{\mu_{1}\nu_{1}\mu_{2}\nu_{2}\mu_{3}\nu_{3}}(p_{1},p_{2},p% _{3}) \displaystyle=\int\mbox{\small$\displaystyle\frac{d^{d}\ell}{(2\pi)^{d}}$}% \mbox{\small$\displaystyle\frac{V^{\mu_{1}\nu_{1}}_{T\phi\phi}(\ell-p_{2},\ell% +p_{3})V^{\mu_{2}\nu_{2}}_{T\phi\phi}(\ell,\ell-p_{2})V^{\mu_{3}\nu_{3}}_{T% \phi\phi}(\ell,\ell+p_{3})}{\ell^{2}(\ell-p_{2})^{2}(\ell+p_{3})^{2}}$} \displaystyle W_{S,1}^{\mu_{1}\nu_{1}\mu_{2}\nu_{2}\mu_{3}\nu_{3}}(p_{1},p_{2}% ,p_{3}) \displaystyle=\mbox{\small$\displaystyle\frac{1}{2}$}\int\mbox{\small$% \displaystyle\frac{d^{d}\ell}{(2\pi)^{d}}$}\mbox{\small$\displaystyle\frac{V^{% \mu_{1}\nu_{1}}_{T\phi\phi}(\ell,\ell+p_{1})V^{\mu_{2}\nu_{2}\mu_{3}\nu_{3}}_{% TT\phi\phi}(\ell,\ell+p_{1})}{\ell^{2}(\ell+p_{1})^{2}}$} \displaystyle W_{S,2}^{\mu_{1}\nu_{1}\mu_{2}\nu_{2}\mu_{3}\nu_{3}}(p_{1},p_{2}% ,p_{3}) \displaystyle=\mbox{\small$\displaystyle\frac{1}{2}$}\int\mbox{\small$% \displaystyle\frac{d^{d}\ell}{(2\pi)^{d}}$}\mbox{\small$\displaystyle\frac{V^{% \mu_{3}\nu_{3}}_{T\phi\phi}(\ell,\ell+p_{3})V^{\mu_{1}\nu_{1}\mu_{2}\nu_{2}}_{% TT\phi\phi}(\ell,\ell+p_{3})}{\ell^{2}(\ell+p_{3})^{2}}$} \displaystyle W_{S,3}^{\mu_{1}\nu_{1}\mu_{2}\nu_{2}\mu_{3}\nu_{3}}(p_{1},p_{2}% ,p_{3}) \displaystyle=\mbox{\small$\displaystyle\frac{1}{2}$}\int\mbox{\small$% \displaystyle\frac{d^{d}\ell}{(2\pi)^{d}}$}\mbox{\small$\displaystyle\frac{V^{% \mu_{2}\nu_{2}}_{T\phi\phi}(\ell,\ell+p_{2})V^{\mu_{1}\nu_{1}\mu_{3}\nu_{3}}_{% TT\phi\phi}(\ell,\ell+p_{2})}{\ell^{2}(\ell+p_{2})^{2}}$} (7.13)

where we have included a symmetry factor 1/2. The calculation of the integral can be simplified by acting with the projectors \Pi on (7.12) in order to write the form factors of the transverse and traceless part of the correlator, as in (5.12)

 \displaystyle\braket{t^{\mu_{1}\nu_{1}}(p_{1})t^{\mu_{2}\nu_{2}}(p_{2})t^{\mu_% {3}\nu_{3}}(p_{3})}_{S} \displaystyle=\,\Pi^{\mu_{1}\nu_{1}}_{\alpha_{1}\beta_{1}}(p_{1})\Pi^{\mu_{2}% \nu_{2}}_{\alpha_{2}\beta_{2}}(p_{2})\Pi^{\mu_{3}\nu_{3}}_{\alpha_{3}\beta_{3}% }(p_{3}) \displaystyle\times\bigg{[}-V_{S}^{\alpha_{1}\beta_{1}\alpha_{2}\beta_{2}% \alpha_{3}\beta_{3}}(p_{1},p_{2},p_{3})+\sum_{i=1}^{3}W_{S,i}^{\alpha_{1}\beta% _{1}\alpha_{2}\beta_{2}\alpha_{3}\beta_{3}}(p_{1},p_{2},p_{3})\bigg{]} (7.14)

7.3 Fermion sector

As in the scalar sector, also in this case we calculate in the one-loop approximation the contribution to the correlation function of the fermion sector by the diagrams in Fig. 3. These contributions can be written as

 \displaystyle\braket{T^{\mu_{1}\nu_{1}}(p_{1})T^{\mu_{2}\nu_{2}}(p_{2})T^{\mu_% {3}\nu_{3}}(p_{3})}_{F}=-\,\sum_{j=1}^{2}V_{F,j}^{\mu_{1}\nu_{1}\mu_{2}\nu_{2}% \mu_{3}\nu_{3}}(p_{1},p_{2},p_{3})+\sum_{j=1}^{3}W_{F,j}^{\mu_{1}\nu_{1}\mu_{2% }\nu_{2}\mu_{3}\nu_{3}}(p_{1},p_{2},p_{3}) (7.15)

using notations similar to the scalar case. In this case we take into account two possible orientations for the fermion in the loop.

Explicitly the terms in (7.15) are given by

 \displaystyle V_{F,1}^{\mu_{1}\nu_{1}\mu_{2}\nu_{2}\mu_{3}\nu_{3}}(p_{1},p_{2}% ,p_{3}) \displaystyle=-\int\mbox{\small$\displaystyle\frac{d^{d}\ell}{(2\pi)^{d}}$}% \mbox{\small$\displaystyle\frac{\text{Tr}\big{[}V^{\mu_{1}\nu_{1}}_{T\bar{\psi% }\psi}(\ell+p_{3},\ell-p_{2})(\not{\ell}+\not{p}_{3})V^{\mu_{3}\nu_{3}}_{T\bar% {\psi}\psi}(\ell,\ell+p_{3})\not{\ell}V^{\mu_{2}\nu_{2}}_{T\bar{\psi}\psi}(% \ell-p_{2},\ell)(\not{\ell}-\not{p}_{2})\big{]}}{\ell^{2}(\ell-p_{2})^{2}(\ell% +p_{3})^{2}}$} \displaystyle V_{F,2}^{\mu_{1}\nu_{1}\mu_{2}\nu_{2}\mu_{3}\nu_{3}}(p_{1},p_{2}% ,p_{3}) \displaystyle=V_{F,1}^{\mu_{1}\nu_{1}\mu_{3}\nu_{3}\mu_{2}\nu_{2}}(p_{1},p_{3}% ,p_{2}) (7.16)
 \displaystyle W_{F,1}^{\mu_{1}\nu_{1}\mu_{2}\nu_{2}\mu_{3}\nu_{3}}(p_{1},p_{2}% ,p_{3}) \displaystyle=-\int\mbox{\small$\displaystyle\frac{d^{d}\ell}{(2\pi)^{d}}$}% \mbox{\small$\displaystyle\frac{\text{Tr}\big{[}V^{\mu_{1}\nu_{1}}_{T\bar{\psi% }\psi}(\ell,\ell+p_{1})\ \not{\ell}\ V^{\mu_{2}\nu_{2}\mu_{3}\nu_{3}}_{TT\bar{% \psi}\psi}(\ell+p_{1},\ell)(\not{\ell}+p_{1})\big{]}}{\ell^{2}(\ell+p_{1})^{2}% }$} \displaystyle W_{F,2}^{\mu_{1}\nu_{1}\mu_{2}\nu_{2}\mu_{3}\nu_{3}}(p_{1},p_{2}% ,p_{3}) \displaystyle=W_{F,1}^{\mu_{3}\nu_{3}\mu_{1}\nu_{1}\mu_{2}\nu_{2}}(p_{3},p_{1}% ,p_{2}) \displaystyle W_{F,3}^{\mu_{1}\nu_{1}\mu_{2}\nu_{2}\mu_{3}\nu_{3}}(p_{1},p_{2}% ,p_{3}) \displaystyle=W_{F,1}^{\mu_{2}\nu_{2}\mu_{1}\nu_{1}\mu_{3}\nu_{3}}(p_{2},p_{1}% ,p_{3}). (7.17)

By a direct computation one can verify that the spin part of the two-gravitons and two-fermions vertex does not contribute to the correlation function.
Acting with the projectors transverse and traceless \Pi we obtain the form factors A_{i}, i=1,\dots,5. For instance, in the fermion sector we obtain

 \displaystyle\braket{t^{\mu_{1}\nu_{1}}(p_{1})t^{\mu_{2}\nu_{2}}(p_{2})t^{\mu_% {3}\nu_{3}}(p_{3})}_{F} \displaystyle=\,\Pi^{\mu_{1}\nu_{1}}_{\alpha_{1}\beta_{1}}(p_{1})\Pi^{\mu_{2}% \nu_{2}}_{\alpha_{2}\beta_{2}}(p_{2})\Pi^{\mu_{3}\nu_{3}}_{\alpha_{3}\beta_{3}% }(p_{3}) \displaystyle\times\bigg{[}-\sum_{j=1}^{2}V_{F,j}^{\alpha_{1}\beta_{1}\alpha_{% 2}\beta_{2}\alpha_{3}\beta_{3}}(p_{1},p_{2},p_{3})+\sum_{j=1}^{3}W_{F,j}^{% \alpha_{1}\beta_{1}\alpha_{2}\beta_{2}\alpha_{3}\beta_{3}}(p_{1},p_{2},p_{3})% \bigg{]} (7.18)

Also in this case the number of fermion families is kept arbitrary and we will multiply the result by a constant n_{F} to account for it. It will be essential for matching this contribution to the general non-perturbative one.

8 Comparisons with the conformal solutions in d=3 and d=5

8.1 Normalization of the two point function

In order to investigate the correspondence between the conformal and the perturbative solutions we briefly recall the result for the \braket{TT} correlator, that we will need in order to investigate the match between the general conformal solution and its perturbative realization. Here we start from a general analysis,based on the CFT solution for this correlator, with a specific application to the case of odd spacetime dimensions, where no renormalization is needed. We will come back to the same correlator in a following section, when we will adress the issue of its renormalization in d=4.

The TT is fixed by conformal invariance in coordinate space to take the form

 \displaystyle\langle T^{\mu\nu}(x)\,T^{\alpha\beta}(y)\rangle=\frac{C_{T}}{(x-% y)^{2d}}\,\mathcal{I}^{\mu\nu,\alpha\beta}(x-y)\,, (8.1)

with

 \displaystyle\mathcal{I}^{\mu\nu,\alpha\beta}(s)=I^{\mu\rho}(x-y)I^{\nu\sigma}% (x-y){\epsilon_{T}}^{\rho\sigma,\alpha\beta}\,, (8.2)

where

 \displaystyle I^{\mu\nu}(x)=\delta^{\mu\nu}-2\frac{x^{\mu}x^{\nu}}{x^{2}} (8.3)

and

 \displaystyle{\epsilon_{T}}^{\mu\nu,\alpha\beta}=\frac{1}{2}\,(\delta^{\mu% \alpha}\delta^{\nu\beta}+\delta^{\mu\beta}\delta^{\nu\alpha}\bigl{)}-\frac{1}{% d}\,\delta^{\mu\nu}\delta^{\alpha\beta}. (8.4)

It is not difficult to check that (8.1) can be cast in the form

 \displaystyle\langle T^{\mu\nu}(x)\,T^{\alpha\beta}(y)\rangle=\frac{C_{T}}{4(d% -2)^{2}d(d+1)}\Delta^{(d)\mu\nu\alpha\beta}(\partial)\frac{1}{((x-y)^{2})^{(d-% 2)}} (8.5)

where

 \displaystyle\hat{\Delta}^{(d)\,\mu\nu\alpha\beta}(\partial) \displaystyle= \displaystyle\frac{1}{2}\left(\hat{\Theta}^{\mu\alpha}\hat{\Theta}^{\nu\beta}+% \hat{\Theta}^{\mu\beta}\hat{\Theta}^{\nu\alpha}\right)-\frac{1}{d-1}\hat{% \Theta^{\mu\nu}}\hat{\Theta}^{\alpha\beta}\,,\quad\text{with}\quad\hat{\Theta}% ^{\mu\nu}=\partial^{\mu}\partial^{\nu}-\delta^{\mu\nu}\,\,\raise 0.5pt\hbox{$% \vbox{\hrule\hbox{\vrule height 6.5043pt\kern 6.5043pt\vrule} \hrule}$} \displaystyle\partial_{\mu}\,\hat{\Delta}^{(d)\,\mu\nu\alpha\beta}(\partial) \displaystyle= \displaystyle 0\,,\quad\delta_{\mu\nu}\,\hat{\Delta}^{(d)\,\mu\nu\alpha\beta}(% \partial)=0 (8.6)

for any function on which it acts. Using the representation

 \displaystyle\frac{1}{(x^{2})^{\alpha}} \displaystyle= \displaystyle\equiv C(\alpha)\,\int d^{d}l\,\frac{e^{il\cdot x}}{(l^{2})^{d/2-% \alpha}} \displaystyle C(\alpha) \displaystyle= \displaystyle\frac{1}{4^{\alpha}\,\pi^{d/2}}\frac{\Gamma(d/2-\alpha)}{\Gamma(% \alpha)} (8.7)

it can be re-expressed in the form

 \displaystyle\braket{T^{\mu\nu}(p)T^{\alpha\beta}(-p)} \displaystyle= \displaystyle\int d^{d}xe^{ip\cdot x}\braket{T^{\mu\nu}(x)T^{\alpha\beta}(0)} (8.8) \displaystyle= \displaystyle C_{T}\frac{\pi^{d/2}\Gamma(-d/2)}{2^{d}(d-2)(d+1)\Gamma(d-2)}p^{% d}\Pi^{\mu\nu\alpha\beta}(p).

Using the expression of the scalar (Euclidean) 2-point function

 {B}_{0}(p_{1}^{2})=\mbox{\small$\displaystyle\frac{1}{\pi^{\frac{d}{2}}}$}\int% \,d^{d}\ell\ \frac{l}{\ell^{2}(\ell-p_{1})^{2}}=\frac{\left[\Gamma\left(\frac{% d}{2}-1\right)\right]^{2}\Gamma\left(2-\frac{d}{2}\right)}{\Gamma\left(d-2% \right)(p_{1}^{2})^{2-\frac{d}{2}}} (8.9)

which is divergent for d=2k, k=1,2,3..., it can also be rewritten in the form

 \braket{T^{\mu\nu}(p)T^{\alpha\beta}(-p)}=4C_{T}\left(\frac{\pi}{4}\right)^{d/% 2}\frac{1}{(d-2)^{2}d(d+1)\Gamma(d/2-1)^{2}}\Pi^{\mu\nu\alpha\beta}(p)\,p^{4}B% _{0}(p^{2}). (8.10)

The singular nature of (8.8) in even dimensions emerges in DR from the appearance of the \Gamma(-d/2) factor, which can be regulated by an ordinary shift d\to d-\epsilon. After a redefinition of the constant C_{T}\to c_{T} which absorbs the d-dependent prefactors, it takes the form

 \displaystyle\braket{T^{\mu\nu}(p)T^{\alpha\beta}(-p)}=c_{T}\,\Pi^{\mu\nu% \alpha\beta}(p)\,\Gamma\left(-\frac{d}{2}+\frac{\epsilon}{2}\right)\,p^{d-\epsilon} (8.11)

where the constant c_{T} is regular and arbitrary. Using the Lagrangian realization of the TT in terms of the two free field theory sectors available in odd dimensions, it can be written as

 \displaystyle\braket{T^{\mu_{1}\nu_{1}}(p)T^{\mu_{2}\nu_{2}}(-p)} \displaystyle=\frac{\pi^{\frac{d}{2}}\big{(}n_{S}+2(d-1)n_{F}\big{)}}{4(d-1)(d% +1)}\,\Pi^{\mu_{1}\nu_{1}\mu_{2}\nu_{2}}(p)\,{B}_{0}(p^{2})\,p^{d} \displaystyle=\frac{\pi^{\frac{d}{2}}\,\big{(}n_{S}+2(d-1)n_{F}\big{)}\,d(d-2)% }{16(d+1)(d-1)\Gamma\left(d-2\right)}\,\left[\Gamma\left(\frac{d}{2}-1\right)% \right]^{2}\,\Pi^{\mu_{1}\nu_{1}\mu_{2}\nu_{2}}(p)\,\Gamma\left(-\frac{d}{2}% \right)\,p^{d} (8.12)

with c_{T} matched in odd dimensions (d>1) according to the expression

 c_{T}=\frac{\pi^{\frac{d}{2}}\,\big{(}n_{S}+2(d-1)n_{F}\big{)}\,d(d-2)}{16(d+1% )(d-1)\Gamma\left(d-2\right)}\,\left[\Gamma\left(\frac{d}{2}-1\right)\right]^{% 2}. (8.13)

In even dimension we have a third (gauge) sector available and therefore it will be necessary to extend (8.1) in order to perform a complete matching. We will address its renormalization in Section 10.
In the case of d=3 and d=5 we get

 c_{T}\ \ \mathrel{\lx@stackrel{{\scriptstyle\makebox[0.0pt]{\mbox{\tiny$d=3$}}% }}{{=}}}\ \ \frac{3\,\pi^{\frac{3}{2}}\,\big{(}n_{S}+4n_{F}\big{)}}{32(3+1)}\,% \left[\Gamma\left(\frac{1}{2}\right)\right]^{2}=\frac{3\pi^{\frac{5}{2}}}{128}% \,\big{(}n_{S}+4n_{F}\big{)}\, (8.14)
 c_{T}\ \ \mathrel{\lx@stackrel{{\scriptstyle\makebox[0.0pt]{\mbox{\tiny$d=5$}}% }}{{=}}}\ \ \frac{15\,\pi^{\frac{5}{2}}\,\big{(}n_{S}+2(5-1)n_{F}\big{)}}{64(5% +1)\Gamma\left(5-2\right)}\,\left[\Gamma\left(\frac{3}{2}\right)\right]^{2}=% \frac{5\pi^{\frac{7}{2}}}{1024}\,\big{(}n_{S}+8n_{F}\big{)}\, (8.15)

We will be using these two expressions of c_{T} in order to perform a comparison with the result of the transverse traceless A_{i} given in [25] for odd dimensions. In such specific cases there are simplifications both from the exact and the perturbative solutions. In particular, the exact solutions turn into rational functions of the momenta, and, as we are going to show, they can be matched with the perturbative ones that we present below.
In d=5, as for all odd dimensions larger than 3, the general solution involves 3 independent constant, as we have mentioned, and we are short of 1 sector in order to match the general result. Neverthless it is still possible to perform a matching between the two solutions, even though not in the most general case.
Clearly, in this case the perturbative results given below for d=5 correspond to a specific choice of the 3 constants of the solution of the conformal constraints. This, obviously, leaves open the issue whether arbitrary choices of all the three constants which appear in odd spacetime dimensions in the general solution correspond to a unitary theory or not, or whether it is possible to formulate, for odd values of values of d>3, CFT’s which do not have a free field theory realization. These may correspond to interacting CFT’s.

8.3 d=3 case

In d=3 the scalar integrals {B}_{0} and {C}_{0} can be computed in a very simple way. In fact in d=3 we get

 {B}_{0}(p_{1}^{2})=\frac{\pi^{3/2}}{\,p_{1}} (8.16)

where p_{1}=|p_{1}|=\sqrt{p_{1}^{2}}, and analogous relations hold for p_{2} and p_{3}. The explicit expression of {C}_{0} can be obtained using the star-triangle relation for which

 \int\frac{d^{d}x}{[(x-x_{1})^{2}]^{\alpha_{1}}\,[(x-x_{2})^{2}]^{\alpha_{2}}\,% [(x-x_{3})^{2}]^{\alpha_{3}}}\ \ \mathrel{\lx@stackrel{{\scriptstyle\makebox[0% .0pt]{\mbox{\tiny$\sum_{i}\alpha_{i}=d$}}}}{{=}}}\ \ \frac{i\pi^{d/2}\nu(% \alpha_{1})\nu(\alpha_{2})\nu(\alpha_{3})}{[(x_{2}-x_{3})^{2}]^{\frac{d}{2}-% \alpha_{1}}\,[(x_{1}-x_{2})^{2}]^{\frac{d}{2}-\alpha_{3}}\,[(x_{1}-x_{3})^{2}]% ^{\frac{d}{2}-\alpha_{2}}} (8.17)

where

 \nu(x)=\frac{\Gamma\left(\frac{d}{2}-x\right)}{\Gamma(x)} (8.18)

that holds only if the condition \sum_{i}\alpha_{i}=d is satisfied. In the case d=3 the LHS of (8.17) is proportional to the three point scalar integral, and in particular

 \displaystyle{C}_{0}(p_{1}^{2},p_{2}^{2},p_{3}^{2}) \displaystyle=\int\,\frac{d^{d}\ell}{\pi^{\frac{d}{2}}}\frac{1}{\ell^{2}(\ell-% p_{2})^{2}(\ell+p_{3})^{2}}=\int\,\frac{d^{d}k}{\pi^{\frac{d}{2}}}\frac{1}{(k-% p_{1})^{2}(k+p_{3})^{2}(k+p_{3}-p_{2})^{2}} \displaystyle=\frac{\left[\Gamma\left(\frac{d}{2}-1\right)\right]^{3}}{\,(p_{1% }^{2})^{\frac{D}{2}-1}(p_{2}^{2})^{\frac{d}{2}-1}(p_{3}^{2})^{\frac{d}{2}-1}}% \ \mathrel{\lx@stackrel{{\scriptstyle\makebox[0.0pt]{\mbox{\tiny$d=3$}}}}{{=}}% }\ \frac{\pi^{3/2}}{\,p_{1}\,p_{2}\,p_{3}}. (8.19)

The explicit expression of the form factors in d=3, using the perturbative approach to one loop order, can be obtained by taking the limit d\to 3 of (8.16) and (8.19) derived from the general diagrammatic expansion. We obtain

 \displaystyle A_{1}^{d=3}(p_{1},p_{2},p_{3}) \displaystyle=\frac{\pi^{3}(n_{S}-4n_{F})}{60(p_{1}+p_{2}+p_{3})^{6}}\Big{[}p_% {1}^{3}+6p_{1}^{2}(p_{3}+p_{2})+(6p_{1}+p_{2}+p_{3})\big{(}(p_{2}+p_{3})^{2}+3% p_{2}p_{3}\big{)}\Big{]} (8.20)
 \displaystyle A_{2}^{d=3}(p_{1},p_{2},p_{3}) \displaystyle=\frac{\pi^{3}(n_{S}-4n_{F})}{60(p_{1}+p_{2}+p_{3})^{6}}\Big{[}4p% _{3}^{2}\big{(}7(p_{1}+p_{2})^{2}+6p_{1}p_{2}\big{)}+20p_{3}^{3}(p_{1}+p_{2})+% 4p_{3}^{4} \displaystyle                                      +3(5p_{3}+p_{1}+p_{2})(p_{1% }+p_{2})\big{(}(p_{1}+p_{2})^{2}+p_{1}p_{2}\big{)}\Big{]} \displaystyle+\frac{\pi^{3}\,n_{F}}{3(p_{1}+p_{2}+p_{3})^{4}}\Big{[}p_{1}^{3}+% 4p_{1}^{2}(p_{2}+p_{3})+(4p_{1}+p_{2}+p_{3})\big{(}(p_{2}+p_{3})^{2}+p_{2}p_{3% }\big{)}\Big{]}
 \displaystyle A_{3}^{d=3}(p_{1},p_{2},p_{3}) \displaystyle=\frac{\pi^{3}(n_{S}-4n_{F})\,p_{3}^{2}}{240(p_{1}+p_{2}+p_{3})^{% 4}}\Big{[}28p_{3}^{2}(p_{1}+p_{2})+3p_{3}\big{(}11(p_{1}+p_{2})^{2}+6p_{1}\,p_% {2}\big{)}+7p_{3}^{3} \displaystyle+12(p_{1}+p_{2})\big{(}(p_{1}+p_{2})^{2}+p_{1}p_{2}\big{)}\Big{]} \displaystyle+\frac{\pi^{3}n_{F}\,p_{3}^{2}}{6(p_{1}+p_{2}+p_{3})^{3}}\Big{[}3% p_{2}(p_{1}+p_{2})+2\big{(}(p_{1}+p_{2})^{2}+p_{1}p_{2}\big{)}+p_{3}^{2}\Big{]} \displaystyle-\frac{\pi^{3}(n_{s}+4n_{F})}{16(p_{1}+p_{2}+p_{3})^{2}}\Big{[}p_% {1}^{3}+2p_{1}^{2}(p_{2}+p_{3})+(2p_{1}+p_{2}+p_{3})\big{(}(p_{2}+p_{3})^{2}-p% _{2}p_{3}\big{)}\Big{]} (8.22)
 \displaystyle A_{4}^{d=3}(p_{1},p_{2},p_{3}) \displaystyle=\frac{\pi^{3}(n_{S}-4n_{F})}{120(p_{1}+p_{2}+p_{3})^{4}}\Big{[}(% 4p_{3}+p_{1}+p_{2})\big{(}3(p_{1}+p_{2})^{4}-3(p_{1}+p_{2})^{2}p_{1}p_{2}+4p_{% 1}^{2}p_{2}^{2}\big{)} \displaystyle+9p_{3}^{2}(p_{1}+p_{2})\big{(}(p_{1}+p_{2})^{2}-3p_{1}p_{2}\big{% )}-3p_{3}^{5}-12p_{3}^{4}(p_{1}+p_{2})-9p_{3}^{3}\big{(}(p_{1}+p_{2})^{2}+2p_{% 1}p_{2}\big{)}\Big{]} \displaystyle+\frac{\pi^{3}\,n_{F}}{6(p_{1}+p_{2}+p_{3})^{3}}\Big{[}(p_{1}+p_{% 2})\big{(}(p_{1}+p_{2})^{2}-p_{1}p_{2}\big{)}(p_{1}+p_{2}+3p_{3})-p_{3}^{4}-3p% _{3}^{3}(p_{1}+p_{2}) \displaystyle-6p_{1}p_{2}p_{3}^{2}\Big{]}-\frac{\pi^{3}(n_{s}+4n_{F})}{8(p_{1}% +p_{2}+p_{3})^{2}}\Big{[}p_{1}^{3}+2p_{1}^{2}(p_{2}+p_{3})+(2p_{1}+p_{2}+p_{3}% )\big{(}(p_{2}+p_{3})^{2}-p_{2}p_{3}\big{)}\Big{]}
 \displaystyle A_{5}^{d=3}(p_{1},p_{2},p_{3}) \displaystyle=\frac{\pi^{3}(n_{S}-4n_{F})}{240(p_{1}+p_{2}+p_{3})^{3}}\Big{[}-% 3(p_{1}+p_{2}+p_{3})^{6}+9(p_{1}+p_{2}+p_{3})^{4}(p_{1}p_{2}+p_{2}p_{3}+p_{1}p% _{3}) \displaystyle+12(p_{1}+p_{2}+p_{3})^{2}(p_{1}p_{2}+p_{2}p_{3}+p_{3}p_{1})^{2}-% 33(p_{1}+p_{2}+p_{3})^{2}p_{1}p_{2}p_{3} \displaystyle+12(p_{1}+p_{2}+p_{3})(p_{1}p_{2}+p_{2}p_{3}+p_{1}p_{3})p_{1}p_{2% }p_{3}+8p_{1}^{2}p_{2}^{2}p_{3}^{2}\Big{]} \displaystyle+\frac{\pi^{3}n_{F}}{12(p_{1}+p_{2}+p_{3})^{2}}\Big{[}-(p_{1}+p_{% 2}+p_{3})^{5}+3(p_{1}+p_{2}+p_{3})^{3}(p_{1}p_{2}+p_{2}p_{3}+p_{1}p_{3}) \displaystyle+4(p_{1}+p_{2}+p_{3})(p_{1}p_{2}+p_{2}p_{3}+p_{1}p_{3})^{2}-11(p_% {1}+p_{2}+p_{3})^{2}p_{1}p_{2}p_{3} \displaystyle+4(p_{1}p_{2}+p_{2}p_{3}+p_{1}p_{3})p_{1}p_{2}p_{3}\Big{]}-\frac{% \pi^{3}(n_{S}+4n_{F})}{16}\Big{[}p_{1}^{3}+p_{2}^{3}+p_{3}^{3}\Big{]} (8.24)

This is in agreement with the expression given in [18] in terms of he constant \alpha_{1},\alpha_{2} and c_{T} if we choose (see [18])

Notice that c_{g} is a constant appearing in [18] related to the possibility of having a nonzero functional variation of the stress energy tensor respect to the metric (\sim\delta T^{\mu\nu}(x)/\delta g_{\alpha\beta}(y)) which is an extra contact term not included in our discussion.

8.4 d=5 case

In this case we have

 {C}_{0}(p_{1}^{2},p_{2}^{2},p_{3}^{2})=\frac{\pi^{3/2}}{p_{1}+p_{2}+p_{3}}. (8.26)

From (8.9) the {B}_{0} is calculated in d=5 as

 {B}_{0}(p_{1}^{2})=-\frac{\pi^{3/2}}{4}p_{1}. (8.27)

In the d\to 5 limit the A_{1} form factor becomes, for instance,

 \displaystyle A_{1}^{d=5}(p_{1},p_{2},p_{3}) \displaystyle=\frac{\pi^{4}(n_{S}-4n_{F})}{560(p_{1}+p_{2}+p_{3})^{7}}\Big{[}(% p_{1}+p_{2}+p_{3})^{2}\big{(}(p_{1}+p_{2}+p_{3})^{4}+(p_{1}+p_{2}+p_{3})^{2}(p% _{1}p_{2}+p_{2}p_{3}+p_{1}p_{3}) \displaystyle+(p_{1}p_{2}+p_{2}p_{3}+p_{1}p_{3})^{2}\big{)}+(p_{1}+p_{2}+p_{3}% )\big{(}(p_{1}+p_{2}+p_{3})^{2}+5(p_{1}p_{2}+p_{2}p_{3}+p_{1}p_{3})\big{)}p_{1% }p_{2}p_{3}+10p_{1}^{2}p_{2}^{2}p_{3}^{2}\Big{]}. (8.28)

The remaining form factors are given in Appendix F. Their expressions are in agreement with those given in [18] when the corresponding constants (denoted by \alpha_{1} and \alpha_{2}) are matched by the relations

The case that we have analysed and their correspondence shows that we can safely move to d=4. In this case we will not attempt a comparison with the results of [18] which are far more involved and require the implementation of some recursion relations on the renormalized 3K integrals. In our case we will have to specialize our computation to d=4 with the inclusion of a third free field theory sector and extract the A_{i}’s after addressing their renormalization.

9 The correlator in d=4 and the trace anomaly

9.1 Gauge and Ghost sectors

We have to consider, as already mentioned, the contributions coming from the spin-1 sector, and in the one-loop approximation they correspond to the diagrams in Fig. 4. We have also to consider the contributions from the ghost, which can be calculated from the same type of diagrams given in Fig. 4 but now with a ghost field running in the loop. A direct computation shows that the ghost and the gauge fixing contributions cancel. Therefore we calculate in the one-loop approximation the contribution to the correlation function of the gauge sector, given by the diagrams in Fig. 4. These contributions can be written as

 \displaystyle\braket{T^{\mu_{1}\nu_{1}}(p_{1})T^{\mu_{2}\nu_{2}}(p_{2})T^{\mu_% {3}\nu_{3}}(p_{3})}_{G}=\,-V_{G}^{\mu_{1}\nu_{1}\mu_{2}\nu_{2}\mu_{3}\nu_{3}}(% p_{1},p_{2},p_{3})+\sum_{i=1}^{3}W_{G,i}^{\mu_{1}\nu_{1}\mu_{2}\nu_{2}\mu_{3}% \nu_{3}}(p_{1},p_{2},p_{3}) (9.1)

for the triangle and the bubble topologies respectively. They are given by

 \displaystyle V_{G}^{\mu_{1}\nu_{1}\mu_{2}\nu_{2}\mu_{3}\nu_{3}}(p_{1},p_{2},p% _{3})= \displaystyle\qquad=\int\mbox{\small$\displaystyle\frac{d^{d}\ell}{(2\pi)^{d}}%$}\mbox{\small$\displaystyle\frac{V^{\mu_{1}\nu_{1}\alpha_{1}\beta_{1}}_{TAA}(% \ell+p_{3},\ell-p_{2})\,\delta_{\alpha_{1}\beta_{2}}\,V^{\mu_{3}\nu_{3}\alpha_% {2}\beta_{2}}_{TAA}(\ell,\ell+p_{3})\,\delta_{\alpha_{2}\beta_{3}}\,V^{\mu_{2}% \nu_{2}\alpha_{3}\beta_{3}}_{TAA}(\ell-p_{2},\ell)\delta_{\alpha_{3}\beta_{1}}% }{\ell^{2}(\ell-p_{2})^{2}(\ell+p_{3})^{2}}$} (9.2)
 \displaystyle W_{G,1}^{\mu_{1}\nu_{1}\mu_{2}\nu_{2}\mu_{3}\nu_{3}}(p_{1},p_{2}% ,p_{3}) \displaystyle=\frac{1}{2}\int\mbox{\small$\displaystyle\frac{d^{d}\ell}{(2\pi)% ^{d}}$}\mbox{\small$\displaystyle\frac{V^{\mu_{1}\nu_{1}\alpha_{1}\beta_{1}}_{% TAA}(\ell,\ell+p_{1})\,\delta_{\beta_{1}\alpha_{2}}\,V^{\mu_{2}\nu_{2}\mu_{3}% \nu_{3}\alpha_{2}\beta_{2}}_{TTAA}(\ell+p_{1},\ell)\,\delta_{\alpha_{1}\beta_{% 2}}}{\ell^{2}(\ell+p_{1})^{2}}$} (9.3) \displaystyle W_{G,2}^{\mu_{1}\nu_{1}\mu_{2}\nu_{2}\mu_{3}\nu_{3}}(p_{1},p_{2}% ,p_{3}) \displaystyle=W_{G,1}^{\mu_{3}\nu_{3}\mu_{1}\nu_{1}\mu_{2}\nu_{2}}(p_{3},p_{1}% ,p_{2}) (9.4) \displaystyle W_{G,3}^{\mu_{1}\nu_{1}\mu_{2}\nu_{2}\mu_{3}\nu_{3}}(p_{1},p_{2}% ,p_{3}) \displaystyle=W_{G,1}^{\mu_{2}\nu_{2}\mu_{1}\nu_{1}\mu_{3}\nu_{3}}(p_{2},p_{1}% ,p_{3}). (9.5)

One can show that the spin part of the two-graviton/two-fermion vertex does not contribute to the correlation function.
By acting with the transverse-traceless \Pi projectors, we obtain the form factors A_{i}, i=1,\dots,5 in the fermion sector, in particular

 \displaystyle\braket{t^{\mu_{1}\nu_{1}}(p_{1})t^{\mu_{2}\nu_{2}}(p_{2})t^{\mu_% {3}\nu_{3}}(p_{3})}_{G} \displaystyle=\,\Pi^{\mu_{1}\nu_{1}}_{\alpha_{1}\beta_{1}}(p_{1})\Pi^{\mu_{2}% \nu_{2}}_{\alpha_{2}\beta_{2}}(p_{2})\Pi^{\mu_{3}\nu_{3}}_{\alpha_{3}\beta_{3}% }(p_{3}) \displaystyle\times\bigg{[}-V_{G}^{\alpha_{1}\beta_{1}\alpha_{2}\beta_{2}% \alpha_{3}\beta_{3}}(p_{1},p_{2},p_{3})+\sum_{i=1}^{3}W_{G,i}^{\alpha_{1}\beta% _{1}\alpha_{2}\beta_{2}\alpha_{3}\beta_{3}}(p_{1},p_{2},p_{3})\bigg{]} (9.6)

Also in this case the number of gauge fields are kept arbitrary by the inclusion of an overall factor n_{G}.

9.2 Divergences

In d=4 the complete correlation function can be written as

 \braket{T^{\mu_{1}\nu_{1}}(p_{1})T^{\mu_{2}\nu_{2}}(p_{2})T^{\mu_{3}\nu_{3}}(p% _{3})}=\sum_{I=F,G,S}\,n_{I}\,\braket{T^{\mu_{1}\nu_{1}}(p_{1})T^{\mu_{2}\nu_{% 2}}(p_{2})T^{\mu_{3}\nu_{3}}(p_{3})}_{I} (9.7)

also valid for the transverse traceless part of the correlator. In this case we encounter divergenes in the forms of single poles in 1/\epsilon (\epsilon=(4-d)/2). In this section we discuss the structures of such divergences and their elimination in DR using the two usual gravitational counterterms.
As a first remark, it is easy to realize for dimensional reasons and power counting that A_{1} is UV finite. All other form factors have divergent parts explicitly given as

 \displaystyle A_{2}^{Div} \displaystyle=\frac{\pi^{2}}{45\,\varepsilon}\,\big{[}26n_{G}-7n_{F}-2n_{S}% \big{]} (9.8a) \displaystyle A_{3}^{Div} \displaystyle=\frac{\pi^{2}}{90\,\varepsilon}\,\big{[}3(s+s_{1})\big{(}6n_{F}+% n_{S}+12n_{G}\big{)}+s_{2}(11n_{F}+62n_{G}+n_{S})\big{]} (9.8b) \displaystyle A_{4}^{Div} \displaystyle=\frac{\pi^{2}}{90\,\varepsilon}\,\big{[}(s+s_{1})\big{(}29n_{F}+% 98n_{G}+4n_{S}\big{)}+s_{2}(43n_{F}+46n_{G}+8n_{S})\big{]} (9.8c) \displaystyle A_{5}^{Div} \displaystyle=\frac{\pi^{2}}{180\,\varepsilon}\bigg{\{}n_{F}\left(43s^{2}-14s(% s_{1}+s_{2})+43s_{1}^{2}-14s_{1}s_{2}+43s_{2}^{2}\right) \displaystyle\quad+2\big{[}n_{G}\left(23s^{2}+26s(s_{1}+s_{2})+23s_{1}^{2}+26s% _{1}s_{2}+23s_{2}^{2}\right)+2n_{S}\left(2s^{2}-s(s_{1}+s_{2})+2s_{1}^{2}-s_{1% }s_{2}+2s_{2}^{2}\right)\big{]}\bigg{\}} (9.8d) .

and at this point we can proceed with their renormalization.

10 Renormalization of the TTT

The renormalization of the 3-graviton vertex is obtained by the addition of 2 counterterms in the defining Lagrangian. In perturbation theory the one loop counterterm Lagrangian is

 S_{count}=-\mbox{\small$\displaystyle\frac{1}{\varepsilon}$}\,\sum_{I=F,S,G}\,% n_{I}\,\int d^{d}x\,\sqrt{-g}\bigg{(}\,\beta_{a}(I)\,C^{2}+\beta_{b}(I)\,E% \bigg{)} (10.1)

corresponding to the Weyl tensor squared and the Euler density, omitting the extra R^{2} operator which is responsible for the \square R term in (3.10), having choosen the local part of anomaly (\sim\beta_{c}\square R) vanishing (\beta_{c}=0). We refer to [28] for a more detailed discussion of this point and of the finite renormalization needed to get from the general \beta_{c}\neq 0 to the \beta_{c}=0 case. The corresponding vertex counterterms are

 \displaystyle\braket{T^{\mu_{1}\nu_{1}}(p_{1})T^{\mu_{2}\nu_{2}}(p_{2})T^{\mu_% {3}\nu_{3}}(p_{3})}_{count}= \displaystyle                 =-\mbox{\small$\displaystyle\frac{1}{\varepsilon% }$}\sum_{I=F,S,G}n_{I}\bigg{(}\beta_{a}(I)\,V_{C^{2}}^{\mu_{1}\nu_{1}\mu_{2}% \nu_{2}\mu_{3}\nu_{3}}(p_{1},p_{2},p_{3})+\beta_{b}(I)\,V_{E}^{\mu_{1}\nu_{1}% \mu_{2}\nu_{2}\mu_{3}\nu_{3}}(p_{1},p_{2},p_{3})\bigg{)} (10.2)

where

 \displaystyle V_{C^{2}}^{\mu_{1}\nu_{1}\mu_{2}\nu_{2}\mu_{3}\nu_{3}}(p_{1},p_{% 2},p_{3}) \displaystyle=8\int\,d^{d}x_{1}\,\,d^{d}x_{2}\,\,d^{d}x_{3}\,\,d^{d}x\,\bigg{(% }\mbox{\small$\displaystyle\frac{\delta^{3}(\sqrt{-g}C^{2})(x)}{\delta g_{\mu_% {1}\nu_{1}}(x_{1})\delta g_{\mu_{2}\nu_{2}}(x_{2})\delta g_{\mu_{3}\nu_{3}}(x_% {3})}$}\bigg{)}_{flat}\,e^{-i(p_{1}\,x_{1}+p_{2}\,x_{2}+p_{3}\,x_{3})} \displaystyle\equiv 8\big{[}\sqrt{-g}\,C^{2}\big{]}^{\mu_{1}\nu_{1}\mu_{2}\nu_% {2}\mu_{3}\nu_{3}}(p_{1},p_{2},p_{3}) (10.3) \displaystyle V_{E}^{\mu_{1}\nu_{1}\mu_{2}\nu_{2}\mu_{3}\nu_{3}}(p_{1},p_{2},p% _{3}) \displaystyle=8\int\,d^{d}x_{1}\,\,d^{d}x_{2}\,\,d^{d}x_{3}\,\,d^{d}x\,\bigg{(% }\mbox{\small$\displaystyle\frac{\delta^{3}(\sqrt{-g}E)(x)}{\delta g_{\mu_{1}% \nu_{1}}(x_{1})\delta g_{\mu_{2}\nu_{2}}(x_{2})\delta g_{\mu_{3}\nu_{3}}(x_{3}% )}$}\bigg{)}_{flat}\,e^{-i(p_{1}\,x_{1}+p_{2}\,x_{2}+p_{3}\,x_{3})} \displaystyle\equiv 8\big{[}\sqrt{-g}\,E\big{]}^{\mu_{1}\nu_{1}\mu_{2}\nu_{2}% \mu_{3}\nu_{3}}(p_{1},p_{2},p_{3}). (10.4)

These vertices satisfy the relations

 \displaystyle\delta_{\mu_{1}\nu_{1}}\,V_{C^{2}}^{\mu_{1}\nu_{1}\mu_{2}\nu_{2}% \mu_{3}\nu_{3}}(p_{1},p_{2},p_{3}) \displaystyle=4(d-4)\big{[}C^{2}\big{]}^{\mu_{2}\nu_{2}\mu_{3}\nu_{3}}(p_{2},p% _{3}) \displaystyle-8\,\bigg{(}[C^{2}]^{\mu_{2}\nu_{2}\mu_{3}\nu_{3}}(p_{1}+p_{2},p_% {3})+[C^{2}]^{\mu_{2}\nu_{2}\mu_{3}\nu_{3}}(p_{2},p_{1}+p_{3})\bigg{)} (10.5) \displaystyle\delta_{\mu_{1}\nu_{1}}\,V_{E}^{\mu_{1}\nu_{1}\mu_{2}\nu_{2}\mu_{% 3}\nu_{3}}(p_{1},p_{2},p_{3}) \displaystyle=4(d-4)\big{[}E\big{]}^{\mu_{2}\nu_{2}\mu_{3}\nu_{3}}(p_{2},p_{3}) (10.6) \displaystyle p_{1\mu_{1}}\,V_{C^{2}}^{\mu_{1}\nu_{1}\mu_{2}\nu_{2}\mu_{3}\nu_% {3}}(p_{1},p_{2},p_{3}) \displaystyle=-4\,\bigg{(}p_{2}^{\nu_{1}}[C^{2}]^{\mu_{2}\nu_{2}\mu_{3}\nu_{3}% }(p_{1}+p_{2},p_{3})+p_{3}^{\nu_{1}}[C^{2}]^{\mu_{2}\nu_{2}\mu_{3}\nu_{3}}(p_{% 2},p_{1}+p_{3})\bigg{)} \displaystyle+4\,p_{2\alpha}\bigg{(}\delta^{\mu_{2}\nu_{1}}[C^{2}]^{\alpha\nu_% {2}\mu_{3}\nu_{3}}(p_{1}+p_{2},p_{3})+\delta^{\nu_{2}\nu_{1}}[C^{2}]^{\alpha% \mu_{2}\mu_{3}\nu_{3}}(p_{1}+p_{2},p_{3})\bigg{)} \displaystyle+4\,p_{3\alpha}\bigg{(}\delta^{\mu_{3}\nu_{1}}[C^{2}]^{\mu_{2}\nu% _{2}\alpha\nu_{3}}(p_{2},p_{1}+p_{3})+\delta^{\nu_{3}\nu_{1}}[C^{2}]^{\mu_{2}% \mu_{2}\mu_{3}\alpha}(p_{2},p_{1}+p_{3})\bigg{)} (10.7) \displaystyle p_{1\mu_{1}}\,V_{E}^{\mu_{1}\nu_{1}\mu_{2}\nu_{2}\mu_{3}\nu_{3}}% (p_{1},p_{2},p_{3}) \displaystyle=0. (10.8)

11 Divergences of the two-point function: a worked out example

Before coming to a discussion of the TTT, in this section we illustrate in some detail the way the generation of the extra tensor structures for such correlators takes place after renormalization. We will work out some of the intermediate steps first of the TT, for simplicity, presenting enough details which will be then applied to the TTT.

We start by extending the analysis of Section 8.1 in the perturbative sector by including all the three sectors (scalar, fermion and gauge) in d dimensions. This choice obviously violates conformal symmetry since the spin 1 contribution is not conformally invariant and it is responsible for an extra trace term proportional to n_{G}. One obtains

 \displaystyle\braket{T^{\mu_{1}\nu_{1}}(p)T^{\mu_{2}\nu_{2}}(-p)} \displaystyle=-\frac{\pi^{2}\,p^{4}}{4(d-1)(d+1)}\,B_{0}(p^{2})\,\Pi^{\mu_{1}% \nu_{1}\mu_{2}\nu_{2}}(p)\Big{[}2(d-1)n_{F}+(2d^{2}-3d-8)n_{G}+n_{S}\Big{]} \displaystyle         +\frac{\pi^{2}\,p^{4}\,n_{G}}{8(d-1)^{2}}(d-4)^{2}(d-2)% \pi^{\mu_{1}\nu_{1}}(p)\pi^{\mu_{2}\nu_{2}}(p)\,B_{0}(p^{2}) (11.1)

with a second contributon proportional to n_{G}. This term vanishes in d=4, as clear from the discussion below.
For this purpose, we recall that around d=4, the projectors are expanded according to the relation

 \Pi^{\,\mu_{1}\nu_{1}\mu_{2}\nu_{2}}(p)=\Pi^{(4)\,\mu_{1}\nu_{1}\mu_{2}\nu_{2}% }(p)-\frac{2}{9}\varepsilon\,\pi^{\mu_{1}\nu_{1}}(p)\,\pi^{\mu_{2}\nu_{2}}(p)+% O(\varepsilon^{2}). (11.2)

This equations requires some clarification and we pause for a moment in order to illustrate its correct use.

A consistent approach to the calculation is to perform all the tensor contractions in d-dimensions and only at the end move to d=4 in the limit of \epsilon\to 0. In this way one is reassured that the contraction of a metric tensor (in this case \delta_{\mu}^{\mu}, being us in the Euclidean case) gives d and not 4. The use of (11.2) is possible only if we are sure that there will not be any trace of the metric to perform. If these conditions are satisfied, then two methods of computation are equivalent and do not generate any ambiguity.
We illustrate this for the TT. Using (11.2) in (11.1), the latter takes the form

 \displaystyle\braket{T^{\mu_{1}\nu_{1}}(p)T^{\mu_{2}\nu_{2}}(-p)} \displaystyle=-\frac{\pi^{2}\,p^{4}}{4}\,\bigg{(}\frac{1}{\varepsilon}+\bar{B}% _{0}(p^{2})\bigg{)}\,\bigg{(}\Pi^{(4)\,\mu_{1}\nu_{1}\mu_{2}\nu_{2}}(p)-\frac{% 2}{9}\varepsilon\,\pi^{\mu_{1}\nu_{1}}(p)\,\pi^{\mu_{2}\nu_{2}}(p)+O(% \varepsilon^{2})\bigg{)} \displaystyle\times\Bigg{[}\bigg{(}\frac{2}{5}+\frac{4}{25}\varepsilon+O(% \varepsilon^{2})\bigg{)}n_{F}+\bigg{(}\frac{4}{5}-\frac{22}{25}\varepsilon+O(% \varepsilon^{2})\bigg{)}n_{G}+\bigg{(}\frac{1}{15}+\frac{16}{225}\varepsilon+O% (\varepsilon^{2})\bigg{)}n_{S}\Bigg{]} \displaystyle+\frac{\pi^{2}\,p^{4}\,n_{G}}{8}\pi^{\mu_{1}\nu_{1}}(p)\pi^{\mu_{% 2}\nu_{2}}(p)\,\bigg{(}\frac{1}{\varepsilon}+\bar{B}_{0}(p^{2})\bigg{)}\bigg{[% }\frac{8}{9}\varepsilon^{2}+\frac{8}{27}\varepsilon^{3}+O(\varepsilon^{4})% \bigg{]} (11.3)

where \Pi^{(4)\,\,\mu_{1}\nu_{1}\mu_{2}\nu_{2}}(p) is the transverse and traceless projector in d=4 and \bar{B}_{0}(p^{2})=2+\log(\mu^{2}/p^{2}) is the finite part in d=4 of the scalar integral in the \overline{MS} scheme. As anticipated above, the last term of (11), generated by the addition of a non-conformal sector (\sim n_{G}) vanishes separately as \epsilon\to 0. Finally, combining all the terms we obtain the regulated (reg) expression of the TT around d=4 in the form

 \displaystyle\braket{T^{\mu_{1}\nu_{1}}(p)T^{\mu_{2}\nu_{2}}(-p)}_{reg} \displaystyle=-\frac{\pi^{2}\,p^{4}}{60\,\varepsilon}\Pi^{(4)\,\mu_{1}\nu_{1}% \mu_{2}\nu_{2}}(p)\left(6n_{F}+12n_{G}+n_{S}\right) \displaystyle+\frac{\pi^{2}\,p^{4}}{270}\pi^{\mu_{1}\nu_{1}}(p)\pi^{\mu_{2}\nu% _{2}}(p)\left(6n_{F}+12n_{G}+n_{S}\right)-\frac{\pi^{2}\,p^{4}}{300}\bar{B}_{0% }(p^{2})\Pi^{\mu_{1}\nu_{1}\mu_{2}\nu_{2}}(p)\left(30n_{F}+60n_{G}+5n_{S}\right) \displaystyle-\frac{\pi^{2}\,p^{4}}{900}\Pi^{\mu_{1}\nu_{1}\mu_{2}\nu_{2}}(p)% \left(36n_{F}-198n_{G}+16n_{S}\right)+O(\varepsilon) (11.4)

The divergence in the previous expression can be removed through the one loop counterterm Lagrangian (10.1). In fact, the second functional derivative of S_{count} with respect to the background metric gives

 \displaystyle\braket{T^{\mu_{1}\nu_{1}}(p)T^{\mu_{2}\nu_{2}}(-p)}_{count} \displaystyle\equiv-\mbox{\small$\displaystyle\frac{1}{\varepsilon}$}\sum_{I=F% ,S,G}\bigg{(}4\beta_{a}(I)\,\big{[}\sqrt{-g}\,C^{2}\big{]}^{\mu_{1}\nu_{1}\mu_% {2}\nu_{2}}(p,-p)\bigg{)} \displaystyle=-\frac{8}{\varepsilon}\frac{(d-3)\,}{(d-2)}p^{4}\Pi^{(d)\,\mu_{1% }\nu_{1}\mu_{2}\nu_{2}}(p)\,\bigg{(}n_{S}\,\beta_{a}(S)+n_{F}\,\beta_{a}(F)+n_% {G}\,\beta_{a}(G)\,\bigg{)} (11.5)

having used the relation V_{E}^{\mu_{1}\nu_{1}\mu_{2}\nu_{2}}(p,-p)=0. In particular, expanding around d=4 and using again (11.2) we obtain

 \displaystyle\braket{T^{\mu_{1}\nu_{1}}(p)T^{\mu_{2}\nu_{2}}(-p)}_{count} \displaystyle=-\frac{8\,p^{4}}{\varepsilon}\bigg{(}\Pi^{(4)\,\mu_{1}\nu_{1}\mu% _{2}\nu_{2}}(p)-\frac{2}{9}\varepsilon\,\pi^{\mu_{1}\nu_{1}}(p)\,\pi^{\mu_{2}% \nu_{2}}(p)+O(\varepsilon^{2})\bigg{)}\,\bigg{(}\frac{1}{2}-\frac{\varepsilon}% {2}+O(\varepsilon^{2})\bigg{)} \displaystyle         \times\bigg{(}n_{S}\,\beta_{a}(S)+n_{F}\,\beta_{a}(F)+n_% {G}\,\beta_{a}(G)\,\bigg{)} \displaystyle=-\mbox{\small$\displaystyle\frac{4}{\varepsilon}$}p^{4}\bigg{(}n% _{S}\,\beta_{a}(S)+n_{F}\,\beta_{a}(F)+n_{G}\,\beta_{a}(G)\,\bigg{)}\,\Pi^{(4)% \,\mu_{1}\nu_{1}\mu_{2}\nu_{2}}(p) \displaystyle+4\,p^{4}\bigg{(}n_{S}\,\beta_{a}(S)+n_{F}\,\beta_{a}(F)+n_{G}\,% \beta_{a}(G)\,\bigg{)}\bigg{[}\Pi^{(4)\,\mu_{1}\nu_{1}\mu_{2}\nu_{2}}(p)+\frac% {2}{9}\pi^{\mu_{1}\nu_{1}}(p)\pi^{\mu_{2}\nu_{2}}(p)\bigg{]}+O(\varepsilon) (11.6)

which cancels the divergence arising in the two point function, if one chooses the parameters as in (3.11). The renormalized 2-point function using (3.11) then takes the form

 \displaystyle\braket{T^{\mu_{1}\nu_{1}}(p)T^{\mu_{2}\nu_{2}}(-p)}_{Ren} \displaystyle=\braket{T^{\mu_{1}\nu_{1}}(p)T^{\mu_{2}\nu_{2}}(-p)}+\braket{T^{% \mu_{1}\nu_{1}}(p)T^{\mu_{2}\nu_{2}}(-p)}_{count} \displaystyle=-\frac{\pi^{2}\,p^{4}}{60}\bar{B}_{0}(p^{2})\Pi^{\mu_{1}\nu_{1}% \mu_{2}\nu_{2}}(p)\left(6n_{F}+12n_{G}+n_{S}\right) \displaystyle\quad-\frac{\pi^{2}\,p^{4}}{900}\Pi^{\mu_{1}\nu_{1}\mu_{2}\nu_{2}% }(p)\big{(}126n_{F}-18n_{G}+31n_{S}\big{)} (11.7)

Notice that the choice of the choice \beta_{c}=0 takes us to a final expression which is transverse and traceless. The same choice of parameters \beta_{a},\,\beta_{b} given in (3.11) removes the divergences in the three point function, as we are going to discuss below.

12 Anomalous Conformal Ward Identities in d=4 and free field content

The divergences arising in the form factors in d=4 and their renormalization induce a breaking of the conformal symmetry, thereby generating a set of anamalous CWI’s. In this section we will give the explicit form of the such identities in the presence of a trace anomaly.

12.1 Primary anomalous CWI’s and free field content

The equations for the anomalous primary CWI’s are generated after renormalization, starting from the d- dimensional expressions of the A_{i}’s given in Appendix G. The renormlization procedure will involve only B_{0}. The primary anomalous CWI’s take the form

 \begin{split}&\displaystyle\textup{K}_{13}A^{Ren}_{3}=2A^{Ren}_{2}-\mbox{% \small$\displaystyle\frac{2\pi^{2}}{45}$}\left(7n_{F}-26n_{G}+2n_{S}\right)\\ &\displaystyle\textup{K}_{23}A^{Ren}_{3}=2A^{Ren}_{2}-\mbox{\small$% \displaystyle\frac{2\pi^{2}}{45}$}\left(7n_{F}-26n_{G}+2n_{S}\right)\\ &\displaystyle\textup{K}_{13}A^{Ren}_{4}=-4A^{Ren}_{2}(p_{2}\leftrightarrow p_% {3})+\mbox{\small$\displaystyle\frac{4\pi^{2}}{45}$}\left(7n_{F}-26n_{G}+2n_{S% }\right)\\ &\displaystyle\textup{K}_{23}A^{Ren}_{4}=-4A^{Ren}_{2}(p_{1}\leftrightarrow p_% {3})+\mbox{\small$\displaystyle\frac{4\pi^{2}}{45}$}\left(7n_{F}-26n_{G}+2n_{S% }\right)\\ &\displaystyle\textup{K}_{13}A^{Ren}_{5}=2\left[A^{Ren}_{4}-A^{Ren}_{4}(p_{1}% \leftrightarrow p_{3})\right]-\mbox{\small$\displaystyle\frac{4\pi^{2}}{9}$}(s% -s_{2})\left(5n_{F}+2n_{G}+n_{s}\right)\\ &\displaystyle\textup{K}_{23}A^{Ren}_{5}=2\left[A^{Ren}_{4}-A^{Ren}_{4}(p_{2}% \leftrightarrow p_{3})\right]-\mbox{\small$\displaystyle\frac{4\pi^{2}}{9}$}(s% _{1}-s_{2})\left(5n_{F}+2n_{G}+n_{s}\right)\end{split} (12.1)

where now the differential operators K_{i} take the form

 K_{i}=\frac{\partial^{2}}{\partial p_{i}^{2}}-\frac{3}{p_{i}}\frac{\partial}{% \partial p_{i}}=4s_{i-1}\frac{\partial^{2}}{\partial s_{i-1}^{2}}-4\frac{% \partial}{\partial s_{i-1}},\qquad i=1,2,3 (12.2)

with the identification s_{0}=s. The (p_{1}\leftrightarrow p_{3}) and (p_{2}\leftrightarrow p_{3}) versions of the anomalous Ward identities can be obtained from (12.1). Using the expressions given in the Appendix E, we can identify the corresponding counterterms for the A_{i}, extracted from the transverse traceless parts of the vertices generated by the counterterm Lagrangian in (10.1), obtaining

 \displaystyle A_{2}^{count} \displaystyle=-\frac{16}{\varepsilon}\,\sum_{I=F,S,G}\,n_{I}\,\big{[}\beta_{a}% (I)+\beta_{b}(I)\big{]} \displaystyle A_{3}^{count} \displaystyle=-\frac{8}{\varepsilon}\,\sum_{I=F,S,G}\,n_{I}\,\big{[}s_{2}\,% \beta_{b}(I)-(s+s_{1})\beta_{a}(I)\big{]}+o(\epsilon) \displaystyle A_{4}^{count} \displaystyle=-\frac{8}{\varepsilon}\,\sum_{I=F,S,G}\,n_{I}\,\big{[}(s+s_{1}-s% _{2})\,\beta_{b}(I)-(s+s_{1}+3s_{2})\beta_{a}(I)\big{]}+o(\epsilon) \displaystyle A_{5}^{count} \displaystyle=-\frac{4}{\varepsilon}\,\sum_{I=F,S,G}\,n_{I}\,\big{[}-\big{(}s^% {2}-2s(s_{1}+s_{2})+(s_{1}-s_{2})^{2}\big{)}\,\beta_{b}(I) \displaystyle                          -\big{(}3s^{2}-2s(s_{1}+s_{2})+3s_{1}^{% 2}-2s_{1}s_{2}+3s_{2}^{2}\big{)}\beta_{a}(I)\big{]}+o(\epsilon) (12.3)

In order to cancel the divergences arising from the form factors, we need to choose the coefficient \beta_{b}(I) and \beta_{a}(I) as in (3.11). The renormalized form factors can then be written as

 \displaystyle A_{2}^{Ren} \displaystyle=A_{2}^{Reg} \displaystyle A_{3}^{Ren} \displaystyle=A_{3}^{Reg}-8\,(s+s_{1}+s_{2})\sum_{I=F,S,G}n_{I}\,\beta_{a}(I) \displaystyle A_{4}^{Ren} \displaystyle=A_{4}^{Reg}-16\,(s+s_{1}+s_{2})\sum_{I=F,S,G}n_{I}\,\beta_{a}(I) \displaystyle A_{5}^{Ren} \displaystyle=A_{5}^{Reg}-8\,(s^{2}+s_{1}^{2}+s_{2}^{2})\sum_{I=F,S,G}n_{I}\,% \beta_{a}(I) (12.4)

where with “reg” we indicate those form factors which remain unmodified by the procedure, being finite. Such are A_{1} and A_{2}.

12.2 Secondary anomalous CWI’s from free field theory

The derivation of the secondary anomalous WI" has been discussed within the general formalism in [25] and in the perturbative approach in [7] in the case of the TJJ correlator. The details of this analysis, which has been discussed at length in our previous work [7], also in this case remain similar. We refer to Appendix A for a definiton of the corresponding operators appearing in such equations and to [7]. A lengthy computation gives

 \displaystyle L_{6}A^{Ren}_{1}+RA^{Ren}_{2}-RA^{Ren}_{2}(p_{2}\leftrightarrow p% _{3})=0 \displaystyle L_{4}\,A^{Ren}_{2}+2p_{1}^{2}\,A^{Ren}_{2}+4RA_{3}-2RA^{Ren}_{4}% (p_{1}\leftrightarrow p_{3})=\frac{4\pi^{2}\,p_{1}^{2}}{45}\left(7n_{F}-26n_{G% }+2n_{S}\right) (12.5) \displaystyle L_{4}\,A^{Ren}_{2}(p_{1}\leftrightarrow p_{3})-R\,A^{Ren}_{4}+RA% ^{Ren}_{4}(p_{2}\leftrightarrow p_{3})+2p_{1}^{2}(A^{Ren}_{2}(p_{2}% \leftrightarrow p_{3})-A^{Ren}_{2})=\frac{2\pi^{2}\,p_{1}^{2}}{45}\left(7n_{F}% -26n_{G}+2n_{S}\right) \displaystyle L_{4}\,A^{Ren}_{2}(p_{2}\leftrightarrow p_{3})-4R\,A^{Ren}_{3}(p% _{2}\leftrightarrow p_{3})+2RA^{Ren}_{4}(p_{1}\leftrightarrow p_{3})-2p_{1}^{2% }A^{Ren}_{2}(p_{2}\leftrightarrow p_{3})=0 \displaystyle L_{2}\,A^{Ren}_{3}(p_{1}\leftrightarrow p_{3})+p_{1}^{2}(A^{Ren}% _{4}-A^{Ren}_{4}(p_{2}\leftrightarrow p_{3})=\frac{30\pi^{2}}{225}(6n_{F}+12n_% {G}+n_{S})\big{(}s_{2}^{2}\bar{B}_{0}(s_{2})-s_{1}^{2}B_{0}^{Reg}(s_{1})\big{)}
 \displaystyle         -\frac{\pi^{2}}{225}\bigg{[}n_{F}\big{(}55s^{2}+5s(29s_{% 1}+7s_{2})+252(s_{1}^{2}-s_{2}^{2})\big{)}+2n_{G}\big{(}155s^{2}+245s\,s_{1}-6% 5s\,s_{2}-18(s_{1}^{2}-s_{2}^{2})\big{)} \displaystyle             +n_{S}\big{(}5s^{2}+10s(2s_{1}+s_{2})+62(s_{1}^{2}-s% _{2}^{2})\big{)}\bigg{]} (12.6) \displaystyle L_{2}\,A^{Ren}_{4}+2R\,A^{Ren}_{5}+8p_{1}^{2}A^{Ren}_{3}(p_{2}% \leftrightarrow p_{3})-2p_{1}^{2}(A^{Ren}_{4}+A^{Ren}_{4}(p_{1}\leftrightarrow p% _{3}))= \displaystyle         -\frac{120\pi^{2}\,s_{1}^{2}}{225}B_{0}(s_{1})\big{(}6n_% {F}+12n_{G}+n_{s}S\big{)}-\frac{4\pi^{2}}{225}\Bigg{[}15s^{2}(6n_{F}+12n_{G}+n% _{S})+5s\,s_{1}(11n_{F}+62n_{G}) \displaystyle             +2s_{1}^{2}(126n_{F}-18n_{G}+31n_{S})\bigg{]} (12.7) \displaystyle L_{2}\,A^{Ren}_{4}(p_{2}\leftrightarrow p_{3})-2R\,A^{Ren}_{5}-8% p_{1}^{2}A^{Ren}_{3}+2p_{1}^{2}(A^{Ren}_{4}(p_{2}\leftrightarrow p_{3})+A^{Ren% }_{4}(p_{1}\leftrightarrow p_{3}))= \displaystyle         +\frac{2\pi^{2}}{225}\bigg{[}60s_{2}^{2}(6n_{F}+12n_{G}+% n_{S})B_{0}^{Reg}(s_{2})+5s\bigg{(}s(7n_{F}-26n_{G}+2n_{S})-s_{1}(43n_{F}+46n_% {G}+8n_{S})\bigg{)} \displaystyle             -5ss_{2}(7n_{F}-26n_{G}+2n_{S})+4s_{2}^{2}(126n_{F}-% 18n_{G}+31n_{S})\bigg{]}. (12.8)

The most involved part of this analysis involves a rewriting of the differential action of the L operators on B_{0} and C_{0}. We have explicitly verified that the renormalized A_{i} satisfy such equations confirming the consistency of the entire approach.

13 Reconstruction of the \braket{TTT} in d=4

In this section we will illustrate the reconstruction procedure for the TTT using the perturbative realization of this correlator. In this case our goal will be to show how the separation of the vertex into a traceless part and an anomaly contribution takes place after renormalization. As already remarked in the introduction, the advantage of using a direct perturbative approach is to present for the transverse traceless sector of this vertex the simplest explicit form, in terms of the renormalized scalar 2- and 3-point functions.
The approach is obviously the standard one, where the renormalization is obtained by the addition to the bare vertex of the counterterms worked out in the previous two sections, but we will try to illustrate in some detail how the generation of the anomaly poles in the trace part takes place in these types of correlators.
We start from the bare local contributions in d dimensions which take the form

 \displaystyle\braket{t_{loc}^{\mu_{1}\nu_{1}}T^{\mu_{2}\nu_{2}}T^{\mu_{3}\nu_{% 3}}} \displaystyle=\Big{(}\mathcal{I}^{\mu_{1}\nu_{1}}_{\alpha_{1}}(p_{1})\,p_{1% \beta_{1}}+\frac{\pi^{\mu_{1}\nu_{1}}(p_{1})}{(d-1)}\delta_{\alpha_{1}\beta_{1% }}\Big{)}\braket{T^{\alpha_{1}\beta_{1}}T^{\mu_{2}\nu_{2}}T^{\mu_{3}\nu_{3}}} \displaystyle=-\frac{2\,\pi^{\mu_{1}\nu_{1}}(p_{1})}{(d-1)}\Big{[}\braket{T^{% \mu_{2}\nu_{2}}(p_{1}+p_{2})T^{\mu_{3}\nu_{3}}(p_{3})}+\braket{T^{\mu_{2}\nu_{% 2}}(p_{2})T^{\mu_{3}\nu_{3}}(p_{1}+p_{3})}\Big{]} \displaystyle+\mathcal{I}^{\mu_{1}\nu_{1}}_{\alpha_{1}}(p_{1})\Big{\{}-p_{2}^{% \alpha_{1}}\braket{T^{\mu_{2}\nu_{2}}(p_{1}+p_{2})T^{\mu_{3}\nu_{3}}(p_{3})}-p% _{3}^{\alpha_{1}}\braket{T^{\mu_{2}\nu_{2}}(p_{2})T^{\mu_{3}\nu_{3}}(p_{1}+p_{% 3})} \displaystyle+p_{2\beta}\Big{[}\delta^{\alpha_{1}\mu_{2}}\braket{T^{\beta\mu_{% 2}}(p_{1}+p_{2})T^{\mu_{3}\nu_{3}}(p_{3})}+\delta^{\alpha_{1}\nu_{2}}\braket{T% ^{\beta\mu_{2}}(p_{1}+p_{2})T^{\mu_{3}\nu_{3}}(p_{3})}\Big{]} \displaystyle+p_{3\beta}\Big{[}\delta^{\alpha_{1}\mu_{3}}\braket{T^{\nu_{2}\mu% _{2}}(p_{2})T^{\beta_{3}\nu_{3}}(p_{1}+p_{3})}+\delta^{\alpha_{1}\nu_{3}}% \braket{T^{\mu_{2}\nu_{2}}(p_{2})T^{\mu_{3}\beta}(p_{1}+p_{3})}\Big{]}\Big{\}} (13.1)

which develop a singularity for \epsilon\to 0, with \epsilon=(4-d)/2, just like all the other contributions appearing in (5.12). We pause for a moment to describe the structure of this expression and comment on the general features of the regularization procedure.
We perform all the tensor contractions in d dimensions and in the final expression we set d=4+\epsilon. For example, if a projector such as \Pi^{(d)} appears, we will be using Eq. (11.2), which relates \Pi^{(d)} to \Pi^{(4)}, and so on. For instance, a projector such as \pi^{\mu_{1}\nu_{1}} with open indices remains unmodified since it has no explicit d-dependence, unless it is contracted with a \delta^{\mu\nu}. It is then clear, from a cursory look at the right hand side of (13) that the regulated expression of this expression involves a prefactor 1/(d-1), which is expanded around d=4 and the replacements of all the two point functions with the regulated expression given by Eq. (11), with the insertion of the appropriate momenta.
The corresponding counterterm is given by

 \displaystyle\braket{t_{loc}^{\mu_{1}\nu_{1}}T^{\mu_{2}\nu_{2}}T^{\mu_{3}\nu_{% 3}}}_{count} \displaystyle=\Big{(}\mathcal{I}^{\mu_{1}\nu_{1}}_{\alpha_{1}}(p_{1})\,p_{1% \beta_{1}}+\frac{\pi^{\mu_{1}\nu_{1}}(p_{1})}{(d-1)}\delta_{\alpha_{1}\beta_{1% }}\Big{)}\braket{T^{\alpha_{1}\beta_{1}}T^{\mu_{2}\nu_{2}}T^{\mu_{3}\nu_{3}}}_% {(count)} \displaystyle=-\frac{1}{\varepsilon}\frac{(d-4)}{(d-1)}\pi^{\mu_{1}\nu_{1}}(p_% {1})\bigg{(}4[E]^{\mu_{2}\nu_{2}\mu_{3}\nu_{3}}(p_{2},p_{3})+4[C^{2}]^{\mu_{2}% \nu_{2}\mu_{3}\nu_{3}}(p_{2},p_{3})\bigg{)} \displaystyle+\frac{1}{\varepsilon}\frac{2}{(d-1)}\pi^{\mu_{1}\nu_{1}}(p_{1})% \bigg{(}4[C^{2}]^{\mu_{2}\nu_{2}\mu_{3}\nu_{3}}(p_{1}+p_{2},p_{3})+4[C^{2}]^{% \mu_{2}\nu_{2}\mu_{3}\nu_{3}}(p_{2},p_{1}+p_{3})\bigg{)} \displaystyle-\frac{1}{\varepsilon}\mathcal{I}^{\mu_{1}\nu_{1}}_{\alpha_{1}}(p% _{1})\bigg{\{}-4p_{2}^{\alpha_{1}}[C^{2}]^{\mu_{2}\nu_{2}\mu_{3}\nu_{3}}(p_{1}% +p_{2},p_{3})-p_{3}^{\alpha_{1}}[C^{2}]^{\mu_{2}\nu_{2}\mu_{3}\nu_{3}}(p_{2},p% _{1}+p_{3}) \displaystyle+4p_{2\beta}\Big{[}\delta^{\alpha_{1}\mu_{2}}[C^{2}]^{\beta\nu_{2% }\mu_{3}\nu_{3}}(p_{1}+p_{2},p_{3})+\delta^{\alpha_{1}\nu_{2}}[C^{2}]^{\mu_{2}% \beta\mu_{3}\nu_{3}}(p_{1}+p_{2},p_{3})\Big{]} \displaystyle+4p_{3\beta}\Big{[}\delta^{\alpha_{1}\mu_{3}}[C^{2}]^{\mu_{2}\nu_% {2}\beta\nu_{3}}(p_{2},p_{1}+p_{3})+\delta^{\alpha_{1}\nu_{3}}[C^{2}]^{\mu_{2}% \nu_{2}\mu_{3}\beta}(p_{2},p_{1}+p_{3})\Big{]}\bigg{\}}. (13.2)

where, for simplicity, we have absorbed the dependence on the total contributions to the beta functions \beta_{a} and \beta_{b}

 \beta_{a,b}\equiv\sum_{I=f,s,G}\beta_{a,b}(I) (13.3)

into [E] and [C^{2}].
It is worth mentioning that all the divergent parts of the local term given in (13) above are cancelled by the local parts of the counterterm (13). For its renormalized expression we obtain

 \displaystyle\braket{t_{loc}^{\mu_{1}\nu_{1}}T^{\mu_{2}\nu_{2}}T^{\mu_{3}\nu_{% 3}}}_{Ren} \displaystyle=\braket{t_{loc}^{\mu_{1}\nu_{1}}T^{\mu_{2}\nu_{2}}T^{\mu_{3}\nu_% {3}}}+\braket{t_{loc}^{\mu_{1}\nu_{1}}T^{\mu_{2}\nu_{2}}T^{\mu_{3}\nu_{3}}}_{(% count)} \displaystyle=\mathcal{V}_{loc\,0\,0}+\braket{t_{loc}^{\mu_{1}\nu_{1}}T^{\mu_{% 2}\nu_{2}}T^{\mu_{3}\nu_{3}}}^{(4)}_{extra} (13.4)

where

with \braket{TT}_{ren} given by (11.7). Notice the presence of an extra contribution coming from the local parts of counterterms that takes the explicit form

 \displaystyle\braket{t_{loc}^{\mu_{1}\nu_{1}}T^{\mu_{2}\nu_{2}}T^{\mu_{3}\nu_{% 3}}}^{(4)}_{extra}=\frac{\hat{\pi}^{\mu_{1}\nu_{1}}(p_{1})}{3\,p_{1}^{2}}\bigg% {(}4[E]^{\mu_{2}\nu_{2}\mu_{3}\nu_{3}}(p_{2},p_{3})+4[C^{2}]^{\mu_{2}\nu_{2}% \mu_{3}\nu_{3}}(p_{2},p_{3})\bigg{)}, (13.6)

having defined

 \displaystyle\hat{\pi}^{\mu\nu}(p)=(\delta^{\mu_{1}\nu_{1}}p^{2}-p^{\mu}p^{\nu}) (13.7)

which shows the emergence of an anomaly pole, similarly to the TJJ cases [4, 5, 6].
The renormalization of the other local contributions follows a similar pattern. In particular, the correlator with two t_{loc} projections takes the form

 \displaystyle\braket{t_{loc}^{\mu_{1}\nu_{1}}t_{loc}^{\mu_{2}\nu_{2}}T^{\mu_{3% }\nu_{3}}}_{Ren}=\mathcal{V}^{\mu_{1}\nu_{1}\mu_{2}\nu_{2}\mu_{3}\nu_{3}}_{loc% \,loc\,0}+\braket{t_{loc}^{\mu_{1}\nu_{1}}t_{loc}^{\mu_{2}\nu_{2}}T^{\mu_{3}% \nu_{3}}}^{(4)}_{extra} (13.8)

where

 \displaystyle\mathcal{V}^{\mu_{1}\nu_{1}\mu_{2}\nu_{2}\mu_{3}\nu_{3}}_{loc\,% loc\,0}=\Big{(}\mathcal{I}^{(4)\,\mu_{2}\nu_{2}}_{\alpha_{2}}(p_{2})\,p_{2% \beta_{2}}+\frac{\pi^{\mu_{2}\nu_{2}}(p_{2})}{3}\delta_{\alpha_{2}\beta_{2}}% \Big{)} \displaystyle\times\Bigg{\{}-\frac{2\,\pi^{\mu_{1}\nu_{1}}(p_{1})}{3}\Big{[}% \braket{T^{\alpha_{2}\beta_{2}}(p_{1}+p_{2})T^{\mu_{3}\nu_{3}}(-p_{1}-p_{2})}_% {Ren}+\braket{T^{\alpha_{2}\beta_{2}}(p_{2})T^{\mu_{3}\nu_{3}}(-p_{2})}_{Ren}% \Big{]} \displaystyle\quad+\mathcal{I}^{(4)\,\mu_{1}\nu_{1}}_{\alpha_{1}}(p_{1})\Big{[% }-p_{2}^{\alpha_{1}}\braket{T^{\alpha_{2}\beta_{2}}(p_{1}+p_{2})T^{\mu_{3}\nu_% {3}}(-p_{1}-p_{2})}_{Ren}-p_{3}^{\alpha_{1}}\braket{T^{\alpha_{2}\beta_{2}}(p_% {2})T^{\mu_{3}\nu_{3}}(-p_{2})}_{Ren} \displaystyle\quad+p_{2\beta}\Big{(}\delta^{\alpha_{1}\alpha_{2}}\braket{T^{% \beta\beta_{2}}(p_{1}+p_{2})T^{\mu_{3}\nu_{3}}(-p_{1}-p_{2})}_{Ren}+\delta^{% \alpha_{1}\beta_{2}}\braket{T^{\beta\alpha_{2}}(p_{1}+p_{2})T^{\mu_{3}\nu_{3}}% (-p_{1}-p_{2})}_{Ren}\Big{)} \displaystyle\quad+p_{3\beta}\Big{(}\delta^{\alpha_{1}\mu_{3}}\braket{T^{\beta% _{2}\alpha_{2}}(p_{2})T^{\beta\nu_{3}}(-p_{2})}_{Ren}+\delta^{\alpha_{1}\nu_{3% }}\braket{T^{\alpha_{2}\beta_{2}}(p_{2})T^{\mu_{3}\beta}(-p_{2})}_{Ren}\Big{)}% \Big{]}\Bigg{\}} (13.9)

in which we define

 \displaystyle\mathcal{I}^{(4)\,\mu\nu}_{\alpha}(p)\equiv\frac{1}{p^{2}}\left[2% p^{(\mu}\delta^{\nu)}_{\alpha}-\frac{p_{\alpha}}{3}\left(\delta^{\mu\nu}+2\,% \frac{p^{\mu}p^{\nu}}{p^{2}}\right)\right] (13.10)

and with the presence of an extra term of the form

 \displaystyle\braket{t_{loc}^{\mu_{1}\nu_{1}}t_{loc}^{\mu_{2}\nu_{2}}T^{\mu_{3% }\nu_{3}}}^{(4)}_{extra}=\frac{\pi^{\mu_{1}\nu_{1}}(p_{1})}{3}\frac{\pi^{\mu_{% 2}\nu_{2}}(p_{2})}{3}\delta_{\alpha_{2}\beta_{2}}\bigg{(}4[E]^{\alpha_{2}\beta% _{2}\mu_{3}\nu_{3}}(p_{2},p_{3})+4[C^{2}]^{\alpha_{2}\beta_{2}\mu_{3}\nu_{3}}(% p_{2},p_{3})\bigg{)} (13.11)

and finally the term with three insertions of t_{loc}

 \displaystyle\braket{t_{loc}^{\mu_{1}\nu_{1}}t_{loc}^{\mu_{2}\nu_{2}}t_{loc}^{% \mu_{3}\nu_{3}}}_{Ren}=\mathcal{V}^{\mu_{1}\nu_{1}\mu_{2}\nu_{2}\mu_{3}\nu_{3}% }_{loc\,loc\,loc}+\braket{t_{loc}^{\mu_{1}\nu_{1}}t_{loc}^{\mu_{2}\nu_{2}}t_{% loc}^{\mu_{3}\nu_{3}}}^{(4)}_{extra} (13.12)

with

 \displaystyle\mathcal{V}^{\mu_{1}\nu_{1}\mu_{2}\nu_{2}\mu_{3}\nu_{3}}_{loc\,% loc\,loc}=\Big{(}\mathcal{I}^{(4)\,\mu_{2}\nu_{2}}_{\alpha_{2}}(p_{2})\,p_{2% \beta_{2}}+\frac{\pi^{\mu_{2}\nu_{2}}(p_{2})}{3}\delta_{\alpha_{2}\beta_{2}}% \Big{)}\Big{(}\mathcal{I}^{(4)\,\mu_{3}\nu_{3}}_{\alpha_{3}}(p_{3})\,p_{3\beta% _{3}}+\frac{\pi^{\mu_{3}\nu_{3}}(p_{3})}{3}\delta_{\alpha_{3}\beta_{3}}\Big{)} \displaystyle\times\Bigg{\{}-\frac{2\,\pi^{\mu_{1}\nu_{1}}(p_{1})}{3}\Big{[}% \braket{T^{\alpha_{2}\beta_{2}}(p_{1}+p_{2})T^{\alpha_{3}\beta_{3}}(-p_{1}-p_{% 2})}_{Ren}+\braket{T^{\alpha_{2}\beta_{2}}(p_{2})T^{\alpha_{3}\beta_{3}}(-p_{2% })}_{Ren}\Big{]} \displaystyle\quad+\mathcal{I}^{(4)\,\mu_{1}\nu_{1}}_{\alpha_{1}}(p_{1})\Big{[% }-p_{2}^{\alpha_{1}}\braket{T^{\alpha_{2}\beta_{2}}(p_{1}+p_{2})T^{\alpha_{3}% \beta_{3}}(-p_{1}-p_{2})}_{Ren}-p_{3}^{\alpha_{1}}\braket{T^{\alpha_{2}\beta_{% 2}}(p_{2})T^{\alpha_{3}\beta_{3}}(-p_{2})}_{Ren} \displaystyle\quad+p_{2\beta}\Big{(}\delta^{\alpha_{1}\alpha_{2}}\braket{T^{% \beta\beta_{2}}(p_{1}+p_{2})T^{\alpha_{3}\beta_{3}}(-p_{1}-p_{2})}_{Ren}+% \delta^{\alpha_{1}\beta_{2}}\braket{T^{\beta\alpha_{2}}(p_{1}+p_{2})T^{\alpha_% {3}\beta_{3}}(-p_{1}-p_{2})}_{Ren}\Big{)} \displaystyle\quad+p_{3\beta}\Big{(}\delta^{\alpha_{1}\alpha_{3}}\braket{T^{% \beta_{2}\alpha_{2}}(p_{2})T^{\beta\beta_{3}}(-p_{2})}_{Ren}+\delta^{\alpha_{1% }\beta_{3}}\braket{T^{\alpha_{2}\beta_{2}}(p_{2})T^{\alpha_{3}\beta}(-p_{2})}_% {Ren}\Big{)}\Big{]}\Bigg{\}} (13.13)

where

 \displaystyle\braket{t_{loc}^{\mu_{1}\nu_{1}}t_{loc}^{\mu_{2}\nu_{2}}t_{loc}^{% \mu_{3}\nu_{3}}}^{(4)}_{extra}=\frac{\pi^{\mu_{1}\nu_{1}}(p_{1})\,\pi^{\mu_{2}% \nu_{2}}(p_{2})\,\pi^{\mu_{3}\nu_{3}}(\bar{p}_{3})}{27}\delta_{\alpha_{2}\beta% _{2}}\delta_{\alpha_{3}\beta_{3}}\bigg{(}4[E]^{\alpha_{2}\beta_{2}\alpha_{3}% \beta_{3}}(p_{2},\bar{p}_{3})+4[C^{2}]^{\alpha_{2}\beta_{2}\alpha_{3}\beta_{3}% }(p_{2},\bar{p}_{3})\bigg{)}. (13.14)

In summary, the counterterm cancels all the divergences arising in the 3-point function and from the local part of the counterterms there are extra contributions in the final renormalized \braket{TTT} of the form

 \displaystyle\braket{T^{\mu_{1}\nu_{1}}T^{\mu_{2}\nu_{2}}T^{\mu_{3}\nu_{3}}}^{% (4)}_{extra} \displaystyle=\left(\frac{\pi^{\mu_{1}\nu_{1}}(p_{1})}{3}\bigg{(}4[E]^{\mu_{2}% \nu_{2}\mu_{3}\nu_{3}}(p_{2},\bar{p}_{3})+4[C^{2}]^{\mu_{2}\nu_{2}\mu_{3}\nu_{% 3}}(p_{2},\bar{p}_{3})\bigg{)}+(\text{perm.})\right) \displaystyle-\left(\frac{\pi^{\mu_{1}\nu_{1}}(p_{1})}{3}\frac{\pi^{\mu_{2}\nu% _{2}}(p_{2})}{3}\delta_{\alpha_{2}\beta_{2}}\bigg{(}4[E]^{\alpha_{2}\beta_{2}% \mu_{3}\nu_{3}}(p_{2},\bar{p}_{3})+4[C^{2}]^{\alpha_{2}\beta_{2}\mu_{3}\nu_{3}% }(p_{2},\bar{p}_{3})\bigg{)}+(\text{perm.})\right) \displaystyle+\frac{\pi^{\mu_{1}\nu_{1}}(p_{1})}{3}\frac{\pi^{\mu_{2}\nu_{2}}(% p_{2})}{3}\frac{\pi^{\mu_{3}\nu_{3}}(\bar{p}_{3})}{3}\delta_{\alpha_{2}\beta_{% 2}}\delta_{\alpha_{3}\beta_{3}}\Big{(}4[E]^{\alpha_{2}\beta_{2}\alpha_{3}\beta% _{3}}(p_{2},\bar{p}_{3})+4[C^{2}]^{\alpha_{2}\beta_{2}\alpha_{3}\beta_{3}}(p_{% 2},\bar{p}_{3})\Big{)}. (13.15)

This extra contribution is exactly the anomalous part of the TTT, which in the flat limit becomes

 \displaystyle\braket{T(p_{1})T^{\mu_{2}\nu_{2}}(p_{2})T^{\mu_{3}\nu_{3}}(\bar{% p}_{3})}_{anomaly}^{(4)} \displaystyle=\big{(}4[E]^{\mu_{2}\nu_{2}\mu_{3}\nu_{3}}(p_{2},p_{3})+4[C^{2}]% ^{\mu_{2}\nu_{2}\mu_{3}\nu_{3}}(p_{2},\bar{p}_{3})\big{)} (13.16) \displaystyle\braket{T(p_{1})T(p_{2})T^{\mu_{3}\nu_{3}}(\bar{p}_{3})}_{anomaly% }^{(4)} \displaystyle=\delta_{\alpha_{2}\beta_{2}}\big{(}4[E]^{\alpha_{2}\beta_{2}\mu_% {3}\nu_{3}}(p_{2},p_{3})+4[C^{2}]^{\alpha_{2}\beta_{2}\mu_{3}\nu_{3}}(p_{2},% \bar{p}_{3})\big{)} \displaystyle\braket{T(p_{1})T(p_{2})T(\bar{p}_{3})}_{anomaly}^{(4)} \displaystyle=\delta_{\alpha_{2}\beta_{2}}\delta_{\alpha_{3}\beta_{3}}\big{(}4% [E]^{\alpha_{2}\beta_{2}\alpha_{3}\beta_{3}}(p_{2},p_{3})+4[C^{2}]^{\alpha_{2}% \beta_{2}\alpha_{3}\beta_{3}}(p_{2},\bar{p}_{3})\big{)} (13.17)

(with T(p)\equiv\delta_{\mu\nu}T^{\mu\nu}). The second order functional derivatives of the anomaly can be reconstructed using the expressions

 \displaystyle\big{[}E\big{]}^{\mu_{i}\nu_{i}\mu_{j}\nu_{j}}(p_{i},p_{j}) \displaystyle=\big{[}R_{\mu\alpha\nu\beta}\,R^{\mu\alpha\nu\beta}\big{]}^{\mu_% {i}\nu_{i}\mu_{j}\nu_{j}}-4\,\big{[}R_{\mu\nu}R^{\mu\nu}\big{]}^{\mu_{i}\nu_{i% }\mu_{j}\nu_{j}}+\big{[}R^{2}\big{]}^{\mu_{i}\nu_{i}\mu_{j}\nu_{j}} \displaystyle=\bigg{\{}\big{[}R_{\mu\alpha\nu\beta}\big{]}^{\mu_{i}\nu_{i}}(p_% {i})\big{[}R^{\mu\alpha\nu\beta}\big{]}^{\mu_{j}\nu_{j}}(p_{j})-4\,\big{[}R_{% \mu\nu}\big{]}^{\mu_{i}\nu_{i}}(p_{i})\big{[}R^{\mu\nu}\big{]}^{\mu_{j}\nu_{j}% }(p_{j})+\big{[}R\big{]}^{\mu_{i}\nu_{i}}(p_{i})\big{[}R\big{]}^{\mu_{j}\nu_{j% }}(p_{j})\bigg{\}} \displaystyle                 +\{(\mu_{i},\nu_{i},p_{i})\leftrightarrow(\mu_{j% },\nu_{j},p_{j})\} (13.18) \displaystyle\big{[}C^{2}\big{]}^{\mu_{i}\nu_{i}\mu_{j}\nu_{j}}(p_{i},p_{j}) \displaystyle=\big{[}R_{\mu\alpha\nu\beta}R^{\mu\alpha\nu\beta}\big{]}^{\mu_{i% }\nu_{i}\mu_{j}\nu_{j}}-2\,\big{[}R_{\mu\nu}R^{\mu\nu}\big{]}^{\mu_{i}\nu_{i}% \mu_{j}\nu_{j}}+\mbox{\small$\displaystyle\frac{1}{3}$}\,\big{[}R^{2}\big{]}^{% \mu_{i}\nu_{i}\mu_{j}\nu_{j}} \displaystyle=\bigg{\{}\big{[}R_{\mu\alpha\nu\beta}\big{]}^{\mu_{i}\nu_{i}}(p_% {i})\big{[}R^{\mu\alpha\nu\beta}\big{]}^{\mu_{j}\nu_{j}}(p_{j})-2\,\big{[}R_{% \mu\nu}\big{]}^{\mu_{i}\nu_{i}}(p_{i})\big{[}R^{\mu\nu}\big{]}^{\mu_{j}\nu_{j}% }(p_{j})+\frac{1}{3}\big{[}R\big{]}^{\mu_{i}\nu_{i}}(p_{i})\big{[}R\big{]}^{% \mu_{j}\nu_{j}}(p_{j})\bigg{\}} \displaystyle                 +\{(\mu_{i},\nu_{i},p_{i})\leftrightarrow(\mu_{j% },\nu_{j},p_{j})\}, (13.19)

for which we obtain

 \displaystyle\braket{T^{\mu_{1}\nu_{1}}(p_{1})T^{\mu_{2}\nu_{2}}(p_{2})T^{\mu_% {3}\nu_{3}}(\bar{p}_{3})}^{(4)}_{extra} \displaystyle=\left(\frac{\pi^{\mu_{1}\nu_{1}}(p_{1})}{3}\braket{T(p_{1})T^{% \mu_{2}\nu_{2}}(p_{2})T^{\mu_{3}\nu_{3}}(\bar{p}_{3})}^{(4)}_{anomaly}+(\text{% perm.})\right) \displaystyle-\left(\frac{\pi^{\mu_{1}\nu_{1}}(p_{1})}{3}\frac{\pi^{\mu_{2}\nu% _{2}}(p_{2})}{3}\braket{T(p_{1})T(p_{2})T^{\mu_{3}\nu_{3}}(\bar{p}_{3})}^{(4)}% _{anomaly}+(\text{perm.})\right) \displaystyle+\frac{\pi^{\mu_{1}\nu_{1}}(p_{1})}{3}\frac{\pi^{\mu_{2}\nu_{2}}(% p_{2})}{3}\frac{\pi^{\mu_{3}\nu_{3}}(\bar{p}_{3})}{3}\braket{T(p_{1})T(p_{2})T% (\bar{p}_{3})}^{(4)}_{anomaly}. (13.20)

13.1 Summary

To summarize, the full renormalized \braket{TTT} in d=4 can be constructed using the renormalized transverse traceless and the local terms. In particular we find

 \displaystyle\braket{T^{\mu_{1}\nu_{1}}(p_{1})T^{\mu_{2}\nu_{2}}(p_{2})T^{\mu_% {3}\nu_{3}}(\bar{p}_{3})}^{(4)}=\braket{t^{\mu_{1}\nu_{1}}(p_{1})t^{\mu_{2}\nu% _{2}}(p_{2})t^{\mu_{3}\nu_{3}}(\bar{p}_{3})}^{(4)}_{Ren} \displaystyle         +\left(\braket{t_{loc}^{\mu_{1}\nu_{1}}(p_{1})T^{\mu_{2}% \nu_{2}}(p_{2})T^{\mu_{3}\nu_{3}}(\bar{p}_{3})}^{(4)}_{Ren}+(\text{cyclic perm% .})\right) \displaystyle-\left(\braket{t_{loc}^{\mu_{1}\nu_{1}}(p_{1})t_{loc}^{\mu_{2}\nu% _{2}}(p_{2})T^{\mu_{3}\nu_{3}}(\bar{p}_{3})}^{(4)}_{Ren}+(\text{cyclic perm.})% \right)+\braket{t_{loc}^{\mu_{1}\nu_{1}}(p_{1})t_{loc}^{\mu_{2}\nu_{2}}(p_{2})% t_{loc}^{\mu_{3}\nu_{3}}(\bar{p}_{3})}^{(4)}_{Ren} (13.21)

where the transverse and traceless parts are expressed as

 \displaystyle\braket{t^{\mu_{1}\nu_{1}}(p_{1})t^{\mu_{2}\nu_{2}}(p_{2})t^{\mu_% {3}\nu_{3}}(\bar{p}_{3})}^{(4)}_{Ren}=\Pi^{(4)\,\mu_{1}\nu_{1}}_{\alpha_{1}% \beta_{1}}(p_{1})\Pi^{(4)\,\mu_{2}\nu_{2}}_{\alpha_{2}\beta_{2}}(p_{2})\Pi^{(4% )\,\mu_{3}\nu_{3}}_{\alpha_{3}\beta_{3}}(\bar{p}_{3}) \displaystyle\times\Big{\{}A_{1}^{Ren}\,p_{2}^{\alpha_{1}}p_{2}^{\beta_{1}}% \bar{p}_{3}^{\alpha_{2}}p_{3}^{\beta_{2}}p_{1}^{\alpha_{3}}p_{1}^{\beta_{3}}+A% _{2}^{Ren}\,\delta^{\beta_{1}\beta_{2}}p_{2}^{\alpha_{1}}p_{3}^{\alpha_{2}}p_{% 1}^{\alpha_{3}}p_{1}^{\beta_{3}}+A_{2}^{ren}\,(p_{1}\leftrightarrow p_{3})\,% \delta^{\beta_{2}\beta_{3}}p_{3}^{\alpha_{2}}p_{1}^{\alpha_{3}}p_{2}^{\alpha_{% 1}}p_{2}^{\beta_{1}} \displaystyle       +A_{2}^{Ren}\,(p_{2}\leftrightarrow p_{3})\,\delta^{\beta_% {3}\beta_{1}}p_{1}^{\alpha_{3}}p_{2}^{\alpha_{1}}p_{3}^{\alpha_{2}}p_{3}^{% \beta_{2}}+A_{3}^{Ren}\,\delta^{\alpha_{1}\alpha_{2}}\delta^{\beta_{1}\beta_{2% }}p_{1}^{\alpha_{3}}p_{1}^{\beta_{3}}+A_{3}^{Ren}(p_{1}\leftrightarrow p_{3})% \,\delta^{\alpha_{2}\alpha_{3}}\delta^{\beta_{2}\beta_{3}}p_{2}^{\alpha_{1}}p_% {2}^{\beta_{1}} \displaystyle           +A_{3}^{Ren}(p_{2}\leftrightarrow p_{3})\,\delta^{% \alpha_{3}\alpha_{1}}\delta^{\beta_{3}\beta_{1}}p_{3}^{\alpha_{2}}p_{3}^{\beta% _{2}}+A_{4}^{Ren}\,\delta^{\alpha_{1}\alpha_{3}}\delta^{\alpha_{2}\beta_{3}}p_% {2}^{\beta_{1}}p_{3}^{\beta_{2}}+A_{4}^{Ren}(p_{1}\leftrightarrow p_{3})\,% \delta^{\alpha_{2}\alpha_{1}}\delta^{\alpha_{3}\beta_{1}}p_{3}^{\beta_{2}}p_{1% }^{\beta_{3}} \displaystyle                              +A_{4}^{Ren}(p_{2}\leftrightarrow p% _{3})\,\delta^{\alpha_{3}\alpha_{2}}\delta^{\alpha_{1}\beta_{2}}p_{1}^{\beta_{% 3}}p_{2}^{\beta_{1}}+A_{5}^{Ren}\delta^{\alpha_{1}\beta_{2}}\delta^{\alpha_{2}% \beta_{3}}\delta^{\alpha_{3}\beta_{1}}\Big{\}} (13.22)

with the renormalized form factors given in Appendix G. It can be further simplified in the form

 \displaystyle\braket{T^{\mu_{1}\nu_{1}}T^{\mu_{2}\nu_{2}}T^{\mu_{3}\nu_{3}}}_{% Ren}=\braket{t^{\mu_{1}\nu_{1}}t^{\mu_{2}\nu_{2}}t^{\mu_{3}\nu_{3}}}_{Ren}+% \braket{T^{\mu_{1}\nu_{1}}T^{\mu_{2}\nu_{2}}T^{\mu_{3}\nu_{3}}}_{Ren\,l\,t}+% \braket{T^{\mu_{1}\nu_{1}}T^{\mu_{2}\nu_{2}}T^{\mu_{3}\nu_{3}}}_{anomaly}

and with the renormalized longitudinal traceless contribution (l\,t ) given by

 \displaystyle\braket{T^{\mu_{1}\nu_{1}}T^{\mu_{2}\nu_{2}}T^{\mu_{3}\nu_{3}}}_{% Ren\,l\,t} \displaystyle\equiv\left(\mathcal{V}^{\mu_{1}\nu_{1}\mu_{2}\nu_{2}\mu_{3}\nu_{% 3}}_{loc\,0\,0}+\mathcal{V}^{\mu_{1}\nu_{1}\mu_{2}\nu_{2}\mu_{3}\nu_{3}}_{0\,% loc\,0}+\mathcal{V}^{\mu_{1}\nu_{1}\mu_{2}\nu_{2}\mu_{3}\nu_{3}}_{0\,0\,loc}\right) \displaystyle-\left(\mathcal{V}^{\mu_{1}\nu_{1}\mu_{2}\nu_{2}\mu_{3}\nu_{3}}_{% loc\,loc\,0}+\mathcal{V}^{\mu_{1}\nu_{1}\mu_{2}\nu_{2}\mu_{3}\nu_{3}}_{0\,loc% \,loc}+\mathcal{V}^{\mu_{1}\nu_{1}\mu_{2}\nu_{2}\mu_{3}\nu_{3}}_{loc\,0\,loc}% \right)+\mathcal{V}^{\mu_{1}\nu_{1}\mu_{2}\nu_{2}\mu_{3}\nu_{3}}_{loc\,loc\,loc} (13.24)

and

 \displaystyle\langle T^{\mu_{1}\nu_{1}}({p}_{1})T^{\mu_{2}\nu_{2}}({p}_{2})T^{% \mu_{3}\nu_{3}}({p}_{3})\rangle_{anomaly}=\frac{\hat{\pi}^{\mu_{1}\nu_{1}}({p}% _{1})}{3\,p_{1}^{2}}\langle T({p}_{1})T^{\mu_{2}\nu_{2}}({p}_{2})T^{\mu_{3}\nu% _{3}}({p}_{3})\rangle_{anomaly} \displaystyle+\frac{\hat{\pi}^{\mu_{2}\nu_{2}}({p}_{2})}{3\,p_{2}^{2}}\langle T% ^{\mu_{1}\nu_{1}}({p}_{1})T({p}_{2})T^{\mu_{3}\nu_{3}}({p}_{3})\rangle_{% anomaly}+\frac{\hat{\pi}^{\mu_{3}\nu_{3}}({p}_{3})}{3\,p_{3}^{2}}\langle T^{% \mu_{1}\nu_{1}}({p}_{1})T^{\mu_{2}\nu_{2}}({p}_{2})T({p}_{3})\rangle_{anomaly} \displaystyle-\>\frac{\hat{\pi}^{\mu_{1}\nu_{1}}({p}_{1})\hat{\pi}^{\mu_{2}\nu% _{2}}({p}_{2})}{9\,p_{1}^{2}p_{2}^{2}}\langle T({p}_{1})T({p}_{2})T^{\mu_{3}% \nu_{3}}({p}_{3})\rangle_{anomaly}-\>\frac{\hat{\pi}^{\mu_{2}\nu_{2}}({p}_{2})% \hat{\pi}^{\mu_{3}\nu_{3}}({p}_{2})}{9p_{2}^{2}p_{3}^{2}}\langle T^{\mu_{1}\nu% _{1}}({p}_{1})T(p_{2})T({p}_{3})\rangle_{anomaly} \displaystyle-\>\frac{\hat{\pi}^{\mu_{1}\nu_{1}}({p}_{1})\hat{\pi}^{\mu_{3}\nu% _{3}}(\bar{p}_{3})}{9p_{1}^{2}p_{3}^{2}}\langle T({p}_{1})T^{\mu_{2}\nu_{2}}({% p}_{2})T({p}_{3})\rangle_{anomaly}+\frac{\hat{\pi}^{\mu_{1}\nu_{1}}({p}_{1})% \hat{\pi}^{\mu_{2}\nu_{2}}({p}_{2})\hat{\pi}^{\mu_{3}\nu_{3}}(\bar{p}_{3})}{27% p_{1}^{2}p_{2}^{2}p_{3}^{2}}\langle T({p}_{1})T({p}_{2})T(\bar{p}_{3})\rangle_% {anomaly}. (13.25)

As a final step, it is convenient to collect together the two renormalized contributions, the transverse and the longitudinal one, which are both traceless, into a single contribution

 \displaystyle\mathcal{V}^{\mu_{1}\nu_{1}\mu_{2}\nu_{2}\mu_{3}\nu_{3}}_{% traceless}\equiv\braket{t^{\mu_{1}\nu_{1}}t^{\mu_{2}\nu_{2}}t^{\mu_{3}\nu_{3}}% }_{Ren}+\braket{T^{\mu_{1}\nu_{1}}T^{\mu_{2}\nu_{2}}T^{\mu_{3}\nu_{3}}}_{Ren\,% l\,t} (13.26)

in order to cast the entire vertex in the form

 \displaystyle\braket{T^{\mu_{1}\nu_{1}}T^{\mu_{2}\nu_{2}}T^{\mu_{3}\nu_{3}}}_{% Ren}=\mathcal{V}^{\mu_{1}\nu_{1}\mu_{2}\nu_{2}\mu_{3}\nu_{3}}_{traceless}+% \braket{T^{\mu_{1}\nu_{1}}T^{\mu_{2}\nu_{2}}T^{\mu_{3}\nu_{3}}}_{anomaly}. (13.27)

We are going to comment briefly on the on the implications of these results at diagrammatic level.

13.2 The perturbative structure of the TTT and the poles separation

The structure of the poles in the TTT is summarized in Fig. 5 where we have denoted with a dashed line the exchange of one or more massless (\sim 1/p_{i}^{2}) interactions. In configuration space such extra terms, related to the renormalization of the correlator, are the natural generalization of the typical anomaly pole interaction found, for instance, in the case of the TJJ, where the effect of the anomaly is in the generation of a nonlocal interaction of the form [4, 5, 21]

 \displaystyle\mathcal{S}_{an}\sim\beta(e)\int d^{4}x\,d^{4}yR^{(1)}(x)\left(% \frac{1}{\square}\right)(x,y)FF(y) (13.28)

with F being the QED field strength and \beta(g) the corresponding beta function of the gauge coupling. In the TTT case, as one can immediately figure out from (13.1) such contributions can be rewritten as contribution to the anomaly action in the form

 \displaystyle\mathcal{S}_{an}\sim\int d^{4}x\,d^{4}yR^{(1)}(x)\left(\frac{1}{% \square}\right)(x,y)\left(\beta_{b}E^{(2)}(y)+\beta_{a}(C^{2})^{(2)}(y)\right) (13.29)

and similar for the other contributions extracted from (13.1). Notice that each \hat{\pi} projector in (13.1) is accompanied by a corresponding anomaly (single) pole of the external invariants, generating contributions of the form 1/p_{i}^{2}, 1/(p_{i}^{2}p_{j}^{2})(i\neq j) and 1/(p_{1}^{2}p_{2}^{2}p_{3}^{2}), where multiple poles are connected to separate external graviton lines. Each momentum invariant appears as a single pole. One can use the correspondence

 \frac{1}{p^{2}}\hat{\pi}^{\mu\nu}\leftrightarrow R^{(1)}\frac{1}{\square} (13.30)

to include such nonzero trace contributions into the anomaly action. This involves a multiplication of the vertex by the external fields together with an integration over all the internal points. As shown in [33] such nonzero trace contributions are automatically generated by the nonlocal conformal anomaly action, which accounts for the entire expression (13.1).
The diagrammatic interpretation suggests a possible generalization of this result also to higher point functions, as one can easily guess, in a combination similar to that shown in Fig. 5.
Notice that the numerators of such decompositions, which correspond to single, double and triple traces are, obviously, purely polynomial in the external invariants, being derived from the anomaly functional, which is local in momentum space.

14 Conclusions and perspectives

We have presented a comparative study of the 3-graviton vertex TTT in CFT’s in momentum space. The analysis of conformal correlators is relatively knew, beyond the standard Lagrangian approach, though the interest in this approach is growing [34, 35]. Our analysis extends a previous work on the TJJ correlator [7] and on the same TTT vertex given in [28], based on similar approaches. Building on the analysis presented in [25, 29], we have gone over the reconstruction program proposed in those works from a perturbative perspective. We have also presented an independent analysis of the solutions of the CWI’s. This is based on a new approach which exploits some properties of the solutions of the hypergeometric systems of differential equations associated with the CWI’s and equivalent to them. The method is alternative to the approach presented in [25], which requires a rather complex analysis of the singularities of the solutions, given in terms of 3-K integrals. The method has been extended by us also to 4-point functions, in the search for special solutions of such systems, which are controlled by a larger class of hypergeometrics, respect to the simpler solutions found for 3-point functions discussed here, as we will show in a forthcoming work.

The comparison with perturbation theory allows to provide drastic simplifications of the results for the vertex, while keeping, for specific dimensions, the generality of the conformal (non-Lagrangian) solution. At the same time, having established a direct link between the two - i.e. the perturbative and the non-Lagrangian formulations - this opens the way to several independent analysis of this vertex - entirely based on the Feynman’s expansion.
This would allow to identify the singularities - and henceforth the anomaly poles - present in such correlator, from a simple and physical perspective based on the analysis of the Landau conditions of the basic (1-loop) diagrams generated by the matching, as done in the simpler case of [22] in a supersymmetric context. The expression that we have presented of such vertex is the simplest one that can be written and down and in d=4 keeps its generality. Obviously, it is possible to extend our analysis to higher (even) dimensions by the inclusion of antisymmetric forms, building on the analysis of [36] as a third (beside scalar and fermions) sector, which would provide an extension of the approach presented in our work.

Finally, we have also discussed the organization of the result fo the TTT - for its renormalized expression - in terms of a homogenous (zero trace) contribution and of an anomaly part. The anomaly (nonzero trace) part, is generated by the renormalization of the local components of the TTT. Our detailed analysis shows that such contributions are not an artifact of the parameterization of the form factors or can be attributed to a specific decomposition but are a general feature of CFT’s and is related to renormalization. This is in agreement with the analysis [29] and, at the same time, with the predictions - limitedly to the anomaly part - coming from the nonlocal anomaly action [33].

Acknowledgements

We thank Emil Mottola for discussion and collaboration on a related project. We thank Kostas Skenderis, Paul McFadden, Fiorenzo Bastianelli, Loriano Bonora, Maxim Chernodub, Olindo Corradini, Luigi Delle Rose and Mirko Serino for discussions. C.C. thanks Fiorenzo Bastianelli and Olindo Corradini for hospitality at the University of Bologna and Giacomo Cacciapaglia and Aldo Deandrea for hospitality at the Codyce 2018 workshop at the University of Lyon. This work is partially supported by INFN Iniziativa specifica QFT-HEP.

Appendix A Secondary CWI’s

The secondary conformal Ward identities are first-order partial differential equations and involve the semi-local information contained in t^{\mu\nu}_{loc}. In order to write them compactly, following [18] we define the two differential operators

 \displaystyle\textup{L}_{N} \displaystyle=p_{1}(p_{1}^{2}+p_{2}^{2}-p_{3}^{2})\frac{\partial}{\partial p_{% 1}}+2p_{1}^{2}p_{2}\frac{\partial}{\partial p_{2}}+\left[(2d-\Delta_{1}-2% \Delta_{2}+N)p_{1}^{2}+(2\Delta_{1}-d)(p_{3}^{2}-p_{2}^{2})\right] (A.1) R \displaystyle=p_{1}\frac{\partial}{\partial p_{1}}-(2\Delta_{1}-d)\,. (A.2)

as well as their symmetric versions

 \displaystyle L^{\prime}_{N}=L_{N},\quad\text{with}\ p_{1}\leftrightarrow p_{2% }\ \text{and}\ \Delta_{1}\leftrightarrow\Delta_{2}, (A.3) \displaystyle R^{\prime}=R,\qquad\text{with}\ p_{1}\mapsto p_{2}\ \text{and}\ % \Delta_{1}\mapsto\Delta_{2}. (A.4)

These operators depend on the conformal dimensions of the operators involved in the 3-point function under consideration, and additionally on a single parameter N determined by the Ward identity in question. In the \braket{TTT} case one finds considering the structure (5.26) one can find for C_{3j}, j=1,\dots,7

 \begin{split}\displaystyle C_{31}&\displaystyle=-\mbox{\small$\displaystyle% \frac{2}{p_{1}^{2}}$}\left[L_{6}A_{1}+RA_{2}-RA_{2}(p_{2}\leftrightarrow p_{3}% )\right]\\ \displaystyle C_{32}&\displaystyle=-\mbox{\small$\displaystyle\frac{1}{p_{1}^{% 2}}$}\left[L_{4}\,A_{2}+2p_{1}^{2}\,A_{2}+4RA_{3}-2RA_{4}(p_{1}\leftrightarrow p% _{3})\right]\\ \displaystyle C_{33}&\displaystyle=-\mbox{\small$\displaystyle\frac{2}{p_{1}^{% 2}}$}\left[L_{4}\,A_{2}(p_{1}\leftrightarrow p_{3})-R\,A_{4}+RA_{4}(p_{2}% \leftrightarrow p_{3})+2p_{1}^{2}(A_{2}(p_{2}\leftrightarrow p_{3})-A_{2})% \right]\\ \displaystyle C_{34}&\displaystyle=-\mbox{\small$\displaystyle\frac{1}{p_{1}^{% 2}}$}\left[L_{4}\,A_{2}(p_{2}\leftrightarrow p_{3})-4R\,A_{3}(p_{2}% \leftrightarrow p_{3})+2RA_{4}(p_{1}\leftrightarrow p_{3})-2p_{1}^{2}A_{2}(p_{% 2}\leftrightarrow p_{3})\right]\\ \displaystyle C_{35}&\displaystyle=-\mbox{\small$\displaystyle\frac{2}{p_{1}^{% 2}}$}\left[L_{2}\,A_{3}(p_{1}\leftrightarrow p_{3})+p_{1}^{2}(A_{4}-A_{4}(p_{2% }\leftrightarrow p_{3})\right]\\ \displaystyle C_{36}&\displaystyle=-\mbox{\small$\displaystyle\frac{1}{p_{1}^{% 2}}$}\left[L_{2}\,A_{4}+2R\,A_{5}+8p_{1}^{2}A_{3}(p_{2}\leftrightarrow p_{3})-% 2p_{1}^{2}(A_{4}+A_{4}(p_{1}\leftrightarrow p_{3}))\right]\\ \displaystyle C_{37}&\displaystyle=-\mbox{\small$\displaystyle\frac{1}{p_{1}^{% 2}}$}\left[L_{2}\,A_{4}(p_{2}\leftrightarrow p_{3})-2R\,A_{5}-8p_{1}^{2}A_{3}+% 2p_{1}^{2}(A_{4}(p_{2}\leftrightarrow p_{3})+A_{4}(p_{1}\leftrightarrow p_{3})% )\right]\\ \end{split} (A.5)

and for C_{4,j}, j=1,\dots,7

 \begin{split}\displaystyle C_{41}&\displaystyle=\mbox{\small$\displaystyle% \frac{2}{p_{2}^{2}}$}\left[L^{\prime}_{6}A_{1}+R^{\prime}A_{2}-R^{\prime}A_{2}% (p_{1}\leftrightarrow p_{3})\right]\\ \displaystyle C_{42}&\displaystyle=\mbox{\small$\displaystyle\frac{1}{p_{2}^{2% }}$}\left[L^{\prime}_{4}\,A_{2}+2p_{2}^{2}\,A_{2}+4R^{\prime}A_{3}-2R^{\prime}% A_{4}(p_{2}\leftrightarrow p_{3})\right]\\ \displaystyle C_{43}&\displaystyle=\mbox{\small$\displaystyle\frac{1}{p_{2}^{2% }}$}\left[L^{\prime}_{4}\,A_{2}(p_{1}\leftrightarrow p_{3})-4R^{\prime}\,A_{3}% (p_{1}\leftrightarrow p_{3})+2RA_{4}(p_{2}\leftrightarrow p_{3})-2p_{2}^{2}A_{% 2}(p_{1}\leftrightarrow p_{3})\right]\\ \displaystyle C_{44}&\displaystyle=\mbox{\small$\displaystyle\frac{2}{p_{2}^{2% }}$}\left[L^{\prime}_{4}\,A_{2}(p_{2}\leftrightarrow p_{3})-R^{\prime}\,A_{4}+% R^{\prime}A_{4}(p_{1}\leftrightarrow p_{3})-2p_{2}^{2}(A_{2}-A_{2}(p_{1}% \leftrightarrow p_{3}))\right]\\ \displaystyle C_{45}&\displaystyle=\mbox{\small$\displaystyle\frac{2}{p_{2}^{2% }}$}\left[L^{\prime}_{2}\,A_{3}(p_{2}\leftrightarrow p_{3})+p_{2}^{2}(A_{4}-A_% {4}(p_{1}\leftrightarrow p_{3})\right]\\ \displaystyle C_{46}&\displaystyle=\mbox{\small$\displaystyle\frac{1}{p_{2}^{2% }}$}\left[L^{\prime}_{2}\,A_{4}+2R^{\prime}\,A_{5}+8p_{2}^{2}A_{3}(p_{1}% \leftrightarrow p_{3})-2p_{2}^{2}(A_{4}+A_{4}(p_{2}\leftrightarrow p_{3}))% \right]\\ \displaystyle C_{47}&\displaystyle=\mbox{\small$\displaystyle\frac{1}{p_{2}^{2% }}$}\left[L^{\prime}_{2}\,A_{4}(p_{1}\leftrightarrow p_{3})-2R^{\prime}\,A_{5}% -8p_{2}^{2}A_{3}+2p_{2}^{2}(A_{4}(p_{2}\leftrightarrow p_{3})+A_{4}(p_{1}% \leftrightarrow p_{3}))\right]\\ \end{split} (A.6)

and finally for C_{5j}, j=1,\dots,7

 \begin{split}\displaystyle C_{51}&\displaystyle=\mbox{\small$\displaystyle% \frac{2}{p_{3}^{2}}$}\bigg{\{}(L_{6}-L^{\prime}_{6})A_{1}+(R+R^{\prime})\big{[% }A_{2}(p_{2}\leftrightarrow p_{3})-A_{2}(p_{1}\leftrightarrow p_{3})\big{]}+2(% d+2)\big{[}A_{2}(p_{2}\leftrightarrow p_{3})-A_{2}(p_{1}\leftrightarrow p_{3})% \big{]}\bigg{\}}\\ \displaystyle C_{52}&\displaystyle=\mbox{\small$\displaystyle\frac{2}{p_{3}^{2% }}$}\bigg{\{}(L_{4}-L^{\prime}_{4})\,A_{2}+2p_{3}^{2}\,\big{[}A_{2}(p_{2}% \leftrightarrow p_{3})-A_{2}(p_{1}\leftrightarrow p_{3})\big{]}\\ &\displaystyle\hskip 56.905512pt+\big{(}R+R^{\prime}+2(d+1)\big{)}\big{[}A_{4}% (p_{1}\leftrightarrow p_{3})-A_{4}(p_{2}\leftrightarrow p_{3})\big{]}\bigg{\}}% \\ \displaystyle C_{53}&\displaystyle=\mbox{\small$\displaystyle\frac{1}{p_{3}^{2% }}$}\bigg{\{}(L_{4}-L^{\prime}_{4})\,A_{2}(p_{1}\leftrightarrow p_{3})+2p_{3}^% {2}A_{2}(p_{1}\leftrightarrow p_{3})-2\big{(}R^{\prime}+R+2(d+1)\big{)}\,\big{% [}2A_{3}(p_{1}\leftrightarrow p_{3})-A_{4}\big{]}\bigg{\}}\\ \displaystyle C_{54}&\displaystyle=\mbox{\small$\displaystyle\frac{1}{p_{3}^{2% }}$}\bigg{\{}(L_{4}-L^{\prime}_{4})\,A_{2}(p_{2}\leftrightarrow p_{3})-2p_{3}^% {2}A_{2}(p_{2}\leftrightarrow p_{3})+2\big{(}R^{\prime}+R+2(d+1)\big{)}\,\big{% [}2A_{3}(p_{2}\leftrightarrow p_{3})-A_{4}\big{]}\bigg{\}}\\ \displaystyle C_{55}&\displaystyle=\mbox{\small$\displaystyle\frac{2}{p_{3}^{2% }}$}\bigg{\{}(L_{2}-L^{\prime}_{2})\,A_{3}-p_{3}^{2}\big{[}A_{4}(p_{2}% \leftrightarrow p_{3})-A_{4}(p_{1}\leftrightarrow p_{3})\big{]}\bigg{\}}\\ \displaystyle C_{56}&\displaystyle=\mbox{\small$\displaystyle\frac{1}{p_{3}^{2% }}$}\bigg{\{}(L_{2}-L^{\prime}_{2})\,A_{4}(p_{1}\leftrightarrow p_{3})+2p_{3}^% {2}\big{[}4A_{3}(p_{2}\leftrightarrow p_{3})-A_{4}-A_{4}(p_{1}\leftrightarrow p% _{3})\big{]}-2\big{(}R+R^{\prime}+2d\big{)}A_{5}\bigg{\}}\\ \displaystyle C_{57}&\displaystyle=\mbox{\small$\displaystyle\frac{1}{p_{3}^{2% }}$}\bigg{\{}(L_{2}-L^{\prime}_{2})\,A_{4}(p_{2}\leftrightarrow p_{3})-2p_{3}^% {2}\big{[}4A_{3}(p_{1}\leftrightarrow p_{3})-A_{4}-A_{4}(p_{2}\leftrightarrow p% _{3})\big{]}+2\big{(}R+R^{\prime}+2d\big{)}A_{5}\bigg{\}}\end{split}

Appendix B Fuchsian solutions of the primary CWI’s

B.1 A_{4} solution

Under the exchange of two momenta, A_{2}(p_{2}\leftrightarrow p_{3}) becomes

 \displaystyle A_{2}(p_{2}\leftrightarrow p_{3}) \displaystyle=p_{3}^{d-4}\sum_{ab}x^{a}y^{\frac{d}{2}-2-a-b}\bigg{[}c^{(2)}(a,% b)\,F_{4}\left(\alpha+2,\beta+2;\gamma,\gamma^{\prime};\mbox{\small$% \displaystyle\frac{x}{y}$},\mbox{\small$\displaystyle\frac{1}{y}$}\right) \displaystyle+\frac{2\,c^{(1)}(a,b)}{\big{(}\beta+2\big{)}}F_{4}(\alpha+3,% \beta+2;\gamma,\gamma^{\prime};\mbox{\small$\displaystyle\frac{x}{y}$},\mbox{% \small$\displaystyle\frac{1}{y}$}\bigg{)}\bigg{]}. (B.1)

In order to solve the equation (6.64) we will be needing the transformation property of F_{4} given by (6.2). Once plugged into the explicit expression on A_{4}(p_{2}\leftrightarrow p_{3}), and separating its 4 indicial components (a_{i},b_{i}) we obtain

 \displaystyle A_{2}(p_{2}\leftrightarrow p_{3}) \displaystyle=p_{3}^{d-4}\Bigg{\{}\ c^{(2)}\left(0,\frac{d}{2}\right)\frac{(d-% 2)}{(d+2)}F_{4}\left(2-\frac{d}{2},2,1-\frac{d}{2},1-\frac{d}{2},x,y\right) \displaystyle                                   +\ c^{(1)}\left(0,\frac{d}{2}% \right)\,\frac{d(d-2)}{(d+2)(d+4)}\,F_{4}\left(2-\frac{d}{2},2,1-\frac{d}{2},-% \frac{d}{2},x,y\right)\Bigg{\}} \displaystyle+p_{3}^{d-4}x^{d/2}\Bigg{\{}\ c^{(2)}\left(\frac{d}{2},0\right)\,% F_{4}\left(\frac{d}{2}+2,2,\frac{d}{2}+1,1-\frac{d}{2},x,y\right)+\ c^{(1)}% \left(\frac{d}{2},0\right)\,\frac{d}{(d+4)}\,F_{4}\left(\frac{d}{2}+2,2,\frac{% d}{2}+1,-\frac{d}{2},x,y\right)\Bigg{\}} \displaystyle+p_{3}^{d-4}y^{d/2}\Bigg{\{}\ c^{(2)}\left(0,0\right)\frac{(d+2)}% {(d-2)}F_{4}\left(2,\frac{d}{2}+2,1-\frac{d}{2},\frac{d}{2}+1,x,y\right) \displaystyle                                   +c^{(1)}\left(0,0\right)\,% \frac{4(d+4)y}{(d-4)(d-2)}F_{4}\left(3,\frac{d}{2}+3,1-\frac{d}{2},\frac{d}{2}% +2,x,y\right)\Bigg{\}} \displaystyle+p_{3}^{d-4}y^{d/2}x^{d/2}(-1)^{d}\Bigg{\{}-\ c^{(2)}\left(\frac{% d}{2},\frac{d}{2}\right)F_{4}\left(d+2,\frac{d}{2}+2,\frac{d}{2}+1,\frac{d}{2}% +1,x,y\right) \displaystyle                                   +\ 4c^{(1)}\left(\frac{d}{2},% \frac{d}{2}\right)\frac{y}{d+2}F_{4}\left(d+3,\frac{d}{2}+3,\frac{d}{2}+1,% \frac{d}{2}+2,x,y\right)\Bigg{\}} (B.2)

and with a similar one for A_{4}(p_{1}\leftrightarrow p_{3}), that we omit. At this stage, using the property of F_{4} given in (6.40) we can rearrange the expression of A_{2}(p_{2}\leftrightarrow p_{3}) as

 \displaystyle A_{2}(p_{2}\leftrightarrow p_{3}) \displaystyle=p_{3}^{d-4}\Bigg{\{}\ c^{(2)}\left(0,\frac{d}{2}\right)\frac{(d-% 2)}{(d+2)}F_{4}\left(2-\frac{d}{2},2,1-\frac{d}{2},1-\frac{d}{2},x,y\right) \displaystyle+\ c^{(1)}\left(0,\frac{d}{2}\right)\,\frac{d(d-2)}{(d+2)(d+4)}\,% F_{4}\left(2-\frac{d}{2},2,1-\frac{d}{2},-\frac{d}{2},x,y\right)\Bigg{\}}+p_{3% }^{d-4}x^{d/2}\Bigg{\{}c^{(2)}\left(\frac{d}{2},0\right) \displaystyle\times\,F_{4}\left(\frac{d}{2}+2,2,\frac{d}{2}+1,1-\frac{d}{2},x,% y\right)+\ c^{(1)}\left(\frac{d}{2},0\right)\,\frac{d}{(d+4)}\,F_{4}\left(% \frac{d}{2}+2,2,\frac{d}{2}+1,-\frac{d}{2},x,y\right)\Bigg{\}} \displaystyle             +p_{3}^{d-4}y^{d/2}\Bigg{\{}\ c^{(2)}\left(0,0\right% )\frac{(d+2)}{(d-2)}F_{4}\left(2,\frac{d}{2}+2,1-\frac{d}{2},\frac{d}{2}+1,x,y\right)
 \displaystyle             +c^{(1)}\left(0,0\right)\,\frac{d(d+2)}{(d-4)(d-2)}% \Bigg{[}F_{4}\left(2,\frac{d}{2}+2,1-\frac{d}{2},\frac{d}{2},x,y\right)-F_{4}% \left(2,\frac{d}{2}+2,1-\frac{d}{2},\frac{d}{2}+1,x,y\right)\Bigg{]}\Bigg{\}} \displaystyle                 +p_{3}^{d-4}y^{d/2}x^{d/2}(-1)^{d}\Bigg{\{}-\ c^% {(2)}\left(\frac{d}{2},\frac{d}{2}\right)F_{4}\left(d+2,\frac{d}{2}+2,\frac{d}% {2}+1,\frac{d}{2}+1,x,y\right) \displaystyle         +\ c^{(1)}\left(\frac{d}{2},\frac{d}{2}\right)\frac{2d}{% (d+4)(d+2)}\Bigg{[}F_{4}\left(d+2,\frac{d}{2}+2,\frac{d}{2}+1,\frac{d}{2},x,y% \right)-F_{4}\left(d+2,\frac{d}{2}+2,\frac{d}{2}+1,\frac{d}{2}+1,x,y\right)% \Bigg{]}\Bigg{\}}. (B.3)

A similar expression can be derived for A_{2}(p_{1}\leftrightarrow p_{3}). The constants appearing in the solution for the form factor A_{4} can be related as

 \displaystyle\begin{aligned} \displaystyle c_{1}^{(4)}\left(0,0\right)&% \displaystyle=\frac{2}{(d+2)}c^{(2)}\left(0,\frac{d}{2}\right)+\frac{2}{(d+2)}% c_{3}^{(4)}\left(0,0\right)\\ \displaystyle c_{1}^{(4)}\left(0,\frac{d}{2}\right)&\displaystyle=-c^{(2)}% \left(0,\frac{d}{2}\right)-\frac{(d+2)}{d}c_{3}^{(4)}\left(\frac{d}{2},0\right% )\\ \displaystyle c_{1}^{(4)}\left(\frac{d}{2},0\right)&\displaystyle=c_{1}^{(4)}% \left(0,\frac{d}{2}\right)\\ \displaystyle c_{1}^{(4)}\left(\frac{d}{2},\frac{d}{2}\right)&\displaystyle=% \frac{2\sec\left(\frac{\pi\,d}{2}\right)\left[\Gamma\left(1-\frac{d}{2}\right)% \right]^{2}}{(d^{2}+6d+8)\Gamma(-d-1)}c^{(1)}\left(0,\frac{d}{2}\right)+\frac{% 2(-1)^{d}}{d+2}c^{(2)}\left(\frac{d}{2},\frac{d}{2}\right)\\ &\displaystyle-\frac{2(d+1)}{d}c_{3}^{(4)}\left(\frac{d}{2},\frac{d}{2}\right)% -\frac{2(-1)^{d/2}\Gamma\left(1-\frac{d}{2}\right)\Gamma\left(-\frac{d}{2}% \right)}{(d+2)\Gamma(-d-1)}c^{(2)}\left(0,\frac{d}{2}\right)\\ \end{aligned}

B.2 Relating the constants in the A_{5} solution

The constants in the expression of A_{5} are fixed as follows

 \displaystyle\begin{aligned} \displaystyle c^{(2)}\left(\frac{d}{2},0\right)&% \displaystyle=c^{(2)}\left(0,\frac{d}{2}\right)\equiv C_{2}\\ \displaystyle c^{(2)}\left(0,0\right)&\displaystyle=\frac{(d-2)}{(d+2)}\,C_{2}% \\ \displaystyle c^{(2)}\left(\frac{d}{2},\frac{d}{2}\right)&\displaystyle=\frac{% \Gamma\left(-\frac{d}{2}\right)\,\Gamma\left(d+2\right)}{\Gamma\left(\frac{d}{% 2}\right)}C_{2}\\ \end{aligned}\hskip 56.905512pt\begin{aligned} \displaystyle c^{(3)}\left(% \frac{d}{2},0\right)&\displaystyle=c^{(3)}\left(0,\frac{d}{2}\right)\equiv C_{% 3}\\ \displaystyle c^{(3)}\left(0,0\right)&\displaystyle=-C_{3}\\ \displaystyle c^{(3)}\left(\frac{d}{2},\frac{d}{2}\right)&\displaystyle=\frac{% \Gamma\left(-\frac{d}{2}\right)\,\Gamma\left(d+1\right)}{\Gamma\left(\frac{d}{% 2}\right)}C_{3}\\ \end{aligned} (B.5)
 \displaystyle\begin{aligned} \displaystyle c^{(5)}\left(\frac{d}{2},0\right)&% \displaystyle=c^{(5)}\left(0,\frac{d}{2}\right)=C_{5}=c^{(5)}\left(0,0\right)% \\ \displaystyle c^{(5)}\left(\frac{d}{2},\frac{d}{2}\right)&\displaystyle=-\frac% {d^{2}\Gamma\left(-\frac{d}{2}-1\right)\Gamma(d+1)}{8\Gamma\left(\frac{d}{2}% \right)}C_{2}+\frac{\Gamma\left(-\frac{d}{2}\right)\,\Gamma(d+1)}{2\Gamma\left% (\frac{d}{2}+1\right)}C_{5}\end{aligned} (B.6)
 \displaystyle\begin{aligned} \displaystyle c_{1}^{(5)}\left(\frac{d}{2},0% \right)&\displaystyle=c_{1}^{(5)}\left(0,\frac{d}{2}\right)=-\frac{d^{2}}{2(d+% 2)(d+4)}C_{1}=c_{1}^{(5)}\left(0,0\right)\\ \displaystyle c_{1}^{(5)}\left(\frac{d}{2},\frac{d}{2}\right)&\displaystyle=-% \frac{d\,\Gamma\left(1-\frac{d}{2}\right)\Gamma(d+1)}{(d+2)(d+4)\Gamma\left(% \frac{d}{2}\right)}C_{1}\end{aligned} (B.7)
 \displaystyle\begin{aligned} \displaystyle c_{2}^{(5)}\left(0,0\right)&% \displaystyle=c_{4}^{(5)}\left(0,0\right)=-\frac{d}{2(d+2)}C_{2}\\ \displaystyle c_{2}^{(5)}\left(\frac{d}{2},0\right)&\displaystyle=c_{4}^{(5)}% \left(0,\frac{d}{2}\right)=\frac{d^{2}}{2(d+2)(d+4)}C_{1}-\frac{d}{d+2}C_{2}\\ \displaystyle c_{2}^{(5)}\left(0,\frac{d}{2}\right)&\displaystyle=c_{4}^{(5)}% \left(\frac{d}{2},0\right)=\frac{d}{d+2}C_{2}\\ \displaystyle c_{2}^{(5)}\left(\frac{d}{2},\frac{d}{2}\right)&\displaystyle=c_% {4}^{(5)}\left(\frac{d}{2},\frac{d}{2}\right)=\frac{d\,\Gamma\left(-\frac{d}{2% }-1\right)\Gamma(d+2)}{4(d+1)\Gamma\left(\frac{d}{2}\right)}C_{2}-\frac{d^{2}% \,\Gamma\left(d+2\right)\Gamma\left(-\frac{d}{2}-1\right)}{4(d+1)(d+4)\,\Gamma% \left(\frac{d}{2}\right)}C_{1}\end{aligned}
 \displaystyle\begin{aligned} \displaystyle c^{(5)}_{3}\left(0,0\right)&% \displaystyle=0\\ \displaystyle c^{(5)}_{3}\left(0,\frac{d}{2}\right)&\displaystyle=c^{(5)}_{3}% \left(\frac{d}{2},0\right)=-C_{2}\frac{d}{d+2}\\ \displaystyle c^{(5)}_{3}\left(\frac{d}{2},\frac{d}{2}\right)&\displaystyle=-% \frac{\pi\,d^{2}\csc\left(\frac{\pi\,d}{2}\right)\Gamma(d+1)}{8\Gamma\left(% \frac{d}{2}+3\right)\Gamma\left(\frac{d}{2}\right)}C_{1}-\frac{\pi\,d(d+4)\csc% \left(\frac{\pi\,d}{2}\right)\Gamma(d+1)}{4\Gamma\left(\frac{d}{2}+3\right)% \Gamma\left(\frac{d}{2}\right)}C_{2}\end{aligned} (B.8)
 \displaystyle\begin{aligned} \displaystyle c_{5}^{(5)}\left(0,0\right)&% \displaystyle=c_{6}^{(5)}\left(0,0\right)=0\\ \displaystyle c_{5}^{(5)}\left(\frac{d}{2},0\right)&\displaystyle=c_{6}^{(5)}% \left(0,\frac{d}{2}\right)=0\\ \displaystyle c_{5}^{(5)}\left(0,\frac{d}{2}\right)&\displaystyle=c_{6}^{(5)}% \left(\frac{d}{2},0\right)=-\frac{d^{2}}{2(d+2)}C_{2}\\ \displaystyle c_{5}^{(5)}\left(\frac{d}{2},\frac{d}{2}\right)&\displaystyle=c_% {6}^{(5)}\left(\frac{d}{2},\frac{d}{2}\right)=\frac{d^{2}\,\Gamma\left(-\frac{% d}{2}-1\right)\,\Gamma(d+1)}{8\,\Gamma\left(\frac{d}{2}\right)}C_{2}\end{aligned}
 \displaystyle\begin{aligned} \displaystyle c_{7}^{(5)}\left(0,0\right)&% \displaystyle=-\frac{d^{2}}{2(d+2)}C_{2}\\ \displaystyle c_{7}^{(5)}\left(\frac{d}{2},0\right)&\displaystyle=c_{7}^{(5)}% \left(0,\frac{d}{2}\right)=\frac{d^{2}}{2(d+2)}C_{2}\\ \displaystyle c_{7}^{(5)}\left(\frac{d}{2},\frac{d}{2}\right)&\displaystyle=-% \frac{d^{2}\,\Gamma\left(-\frac{d}{2}-1\right)\,\Gamma(d+1)}{8\,\Gamma\left(% \frac{d}{2}\right)}C_{2}\end{aligned}

Appendix C Summary: Reconstructions for odd dimensions

We summarize the main steps collecting all the equations needed for the reconstruction in odd dimensions using the matched solutions in d=3 and 5. The vertex is reconstructed from the relation

 \displaystyle\braket{T^{\mu_{1}\nu_{1}}\,T^{\mu_{2}\nu_{2}}\,T^{\mu_{3}\nu_{3}}} \displaystyle=\braket{t^{\mu_{1}\nu_{1}}\,t^{\mu_{2}\nu_{2}}\,t^{\mu_{3}\nu_{3% }}}+\braket{T^{\mu_{1}\nu_{1}}\,T^{\mu_{2}\nu_{2}}\,t_{loc}^{\mu_{3}\nu_{3}}}+% \braket{T^{\mu_{1}\nu_{1}}\,t_{loc}^{\mu_{2}\nu_{2}}\,T^{\mu_{3}\nu_{3}}} \displaystyle+\braket{t_{loc}^{\mu_{1}\nu_{1}}\,T^{\mu_{2}\nu_{2}}\,T^{\mu_{3}% \nu_{3}}}-\braket{T^{\mu_{1}\nu_{1}}\,t_{loc}^{\mu_{2}\nu_{2}}\,t_{loc}^{\mu_{% 3}\nu_{3}}}-\braket{t_{loc}^{\mu_{1}\nu_{1}}\,t_{loc}^{\mu_{2}\nu_{2}}\,T^{\mu% _{3}\nu_{3}}} \displaystyle-\braket{t_{loc}^{\mu_{1}\nu_{1}}\,T^{\mu_{2}\nu_{2}}\,t_{loc}^{% \mu_{3}\nu_{3}}}+\braket{t_{loc}^{\mu_{1}\nu_{1}}\,t_{loc}^{\mu_{2}\nu_{2}}\,t% _{loc}^{\mu_{3}\nu_{3}}} (C.1)

The transverse traceless sector \braket{ttt} is constructed using the corresponsing form factors A_{i} in the respective dimensions using (5.12). Their explicit expressions are found in Section 8.2. For the local terms in (5) we use eq. (13)

 \displaystyle\braket{t_{loc}^{\mu_{1}\nu_{1}}(p_{1})T^{\mu_{2}\nu_{2}}(p_{2})T% ^{\mu_{3}\nu_{3}}(\bar{p}_{3})}=\Big{(}\mathcal{I}^{\mu_{1}\nu_{1}}_{\alpha_{1% }}(p_{1})\,p_{1\beta_{1}}+\frac{\pi^{\mu_{1}\nu_{1}}(p_{1})}{(d-1)}\delta_{% \alpha_{1}\beta_{1}}\Big{)}\braket{T^{\alpha_{1}\beta_{1}}(p_{1})T^{\mu_{2}\nu% _{2}}(p_{2})T^{\mu_{3}\nu_{3}}(\bar{p}_{3})} \displaystyle             =-\frac{2\,\pi^{\mu_{1}\nu_{1}}(p_{1})}{(d-1)}\Big{[% }\braket{T^{\alpha_{2}\beta_{2}}(p_{1}+p_{2})T^{\mu_{3}\nu_{3}}(-p_{1}-p_{2})}% +\braket{T^{\alpha_{2}\beta_{2}}(p_{2})T^{\mu_{3}\nu_{3}}(-p_{2})}\Big{]} \displaystyle                +\mathcal{I}^{\mu_{1}\nu_{1}}_{\alpha_{1}}(p_{1})% \Big{[}-p_{2}^{\alpha_{1}}\braket{T^{\alpha_{2}\beta_{2}}(p_{1}+p_{2})T^{\mu_{% 3}\nu_{3}}(-p_{1}-p_{2})}-p_{3}^{\alpha_{1}}\braket{T^{\alpha_{2}\beta_{2}}(p_% {2})T^{\mu_{3}\nu_{3}}(-p_{2})} \displaystyle                +p_{2\beta}\Big{(}\delta^{\alpha_{1}\alpha_{2}}% \braket{T^{\beta\beta_{2}}(p_{1}+p_{2})T^{\mu_{3}\nu_{3}}(-p_{1}-p_{2})}+% \delta^{\alpha_{1}\beta_{2}}\braket{T^{\beta\alpha_{2}}(p_{1}+p_{2})T^{\mu_{3}% \nu_{3}}(-p_{1}-p_{2})}\Big{)} \displaystyle                +p_{3\beta}\Big{(}\delta^{\alpha_{1}\mu_{3}}% \braket{T^{\beta_{2}\alpha_{2}}(p_{2})T^{\beta\nu_{3}}(-p_{2})}+\delta^{\alpha% _{1}\nu_{3}}\braket{T^{\alpha_{2}\beta_{2}}(p_{2})T^{\mu_{3}\beta}(-p_{2})}% \Big{)}\Big{]} (C.2)

with the two-point functions defined as in

 \braket{T^{\mu\nu}(p)T^{\alpha\beta}(-p)}=c_{T}\,\Pi^{\mu\nu\alpha\beta}(p)\,% \Gamma\left(-\frac{d}{2}+\frac{\epsilon}{2}\right)\,p^{d-\epsilon} (C.3)

with

 c_{T}=\frac{3\pi^{\frac{5}{2}}}{128}\,\big{(}n_{S}+4n_{F}\big{)}\, (C.4)

in d=3 and

 c_{T}=\frac{5\pi^{\frac{7}{2}}}{1024}\,\big{(}n_{S}+8n_{F}\big{)}\, (C.5)

in d=5. In d=3 we generate this way the most general CFT solution which matches the analogous case presented in [18].
The \braket{t_{loc}t_{loc}T} and \braket{t_{loc}t_{loc}t_{loc}} terms are obtained from

 \displaystyle\braket{t_{loc}^{\mu_{1}\nu_{1}}(p_{1})t_{loc}^{\mu_{2}\nu_{2}}(p% _{2})T^{\mu_{3}\nu_{3}}(\bar{p}_{3})}=\Big{(}\mathcal{I}^{\mu_{2}\nu_{2}}_{% \alpha_{2}}(p_{2})\,p_{2\beta_{2}}+\frac{\pi^{\mu_{2}\nu_{2}}(p_{2})}{(d-1)}% \delta_{\alpha_{2}\beta_{2}}\Big{)} \displaystyle             \times\Bigg{\{}-\frac{2\,\pi^{\mu_{1}\nu_{1}}(p_{1})% }{(d-1)}\Big{[}\braket{T^{\alpha_{2}\beta_{2}}(p_{1}+p_{2})T^{\mu_{3}\nu_{3}}(% -p_{1}-p_{2})}+\braket{T^{\alpha_{2}\beta_{2}}(p_{2})T^{\mu_{3}\nu_{3}}(-p_{2}% )}\Big{]} \displaystyle                +\mathcal{I}^{\mu_{1}\nu_{1}}_{\alpha_{1}}(p_{1})% \Big{[}-p_{2}^{\alpha_{1}}\braket{T^{\alpha_{2}\beta_{2}}(p_{1}+p_{2})T^{\mu_{% 3}\nu_{3}}(-p_{1}-p_{2})}-p_{3}^{\alpha_{1}}\braket{T^{\alpha_{2}\beta_{2}}(p_% {2})T^{\mu_{3}\nu_{3}}(-p_{2})} \displaystyle                +p_{2\beta}\Big{(}\delta^{\alpha_{1}\alpha_{2}}% \braket{T^{\beta\beta_{2}}(p_{1}+p_{2})T^{\mu_{3}\nu_{3}}(-p_{1}-p_{2})}+% \delta^{\alpha_{1}\beta_{2}}\braket{T^{\beta\alpha_{2}}(p_{1}+p_{2})T^{\mu_{3}% \nu_{3}}(-p_{1}-p_{2})}\Big{)} \displaystyle                +p_{3\beta}\Big{(}\delta^{\alpha_{1}\mu_{3}}% \braket{T^{\beta_{2}\alpha_{2}}(p_{2})T^{\beta\nu_{3}}(-p_{2})}+\delta^{\alpha% _{1}\nu_{3}}\braket{T^{\alpha_{2}\beta_{2}}(p_{2})T^{\mu_{3}\beta}(-p_{2})}% \Big{)}\Big{]}\Bigg{\}} (C.6)
 \displaystyle\braket{t_{loc}^{\mu_{1}\nu_{1}}(p_{1})t_{loc}^{\mu_{2}\nu_{2}}(p% _{2})t_{loc}^{\mu_{3}\nu_{3}}(\bar{p}_{3})}=\left(\mathcal{I}^{\mu_{3}\nu_{3}}% _{\alpha_{3}}(\bar{p}_{3})\,\bar{p}_{3\beta_{3}}+\frac{\pi^{\mu_{3}\nu_{3}}(% \bar{p}_{3})}{(d-1)}\delta_{\alpha_{3}\beta_{3}}\right)\left(\mathcal{I}^{\mu_% {2}\nu_{2}}_{\alpha_{2}}(p_{2})\,p_{2\beta_{2}}+\frac{\pi^{\mu_{2}\nu_{2}}(p_{% 2})}{(d-1)}\delta_{\alpha_{2}\beta_{2}}\right) \displaystyle             \times\Bigg{\{}-\frac{2\,\pi^{\mu_{1}\nu_{1}}(p_{1})% }{(d-1)}\Big{[}\braket{T^{\alpha_{2}\beta_{2}}(p_{1}+p_{2})T^{\alpha_{3}\beta_% {3}}(-p_{1}-p_{2})}+\braket{T^{\alpha_{2}\beta_{2}}(p_{2})T^{\alpha_{3}\beta_{% 3}}(-p_{2})}\Big{]} \displaystyle             \quad+\mathcal{I}^{\mu_{1}\nu_{1}}_{\alpha_{1}}(p_{1% })\Big{[}-p_{2}^{\alpha_{1}}\braket{T^{\alpha_{2}\beta_{2}}(p_{1}+p_{2})T^{% \alpha_{3}\beta_{3}}(-p_{1}-p_{2})}-p_{3}^{\alpha_{1}}\braket{T^{\alpha_{2}% \beta_{2}}(p_{2})T^{\alpha_{3}\beta_{3}}(-p_{2})} \displaystyle             \quad+p_{2\beta}\Big{(}\delta^{\alpha_{1}\alpha_{2}}% \braket{T^{\beta\beta_{2}}(p_{1}+p_{2})T^{\alpha_{3}\beta_{3}}(-p_{1}-p_{2})}+% \delta^{\alpha_{1}\beta_{2}}\braket{T^{\beta\alpha_{2}}(p_{1}+p_{2})T^{\alpha_% {3}\beta_{3}}(-p_{1}-p_{2})}\Big{)} \displaystyle             \quad+p_{3\beta}\Big{(}\delta^{\alpha_{1}\alpha_{3}}% \braket{T^{\beta_{2}\alpha_{2}}(p_{2})T^{\beta\beta_{3}}(-p_{2})}+\delta^{% \alpha_{1}\beta_{3}}\braket{T^{\alpha_{2}\beta_{2}}(p_{2})T^{\alpha_{3}\beta}(% -p_{2})}\Big{)}\Big{]}\Bigg{\}} (C.7)

by inserting the corresponding values of d. The matching with the analogous solution given in [18] is obtained, in d=3, for

and in d=5 we use similar matching conditions

Appendix D Vertices

We have shown in Fig. 1 a list of all the vertices which are needed for the momentum space computation of the TTT correlator. We list them below and they are computed by taking functional derivatives of the action in order to allows to keep multi-graviton correlators symmetric. We use the letter V to indicate the vertex, the subscript is referred to the fields involved and the Greek indices are linked to the Lorentz structure of the space-time. Furthermore about the momenta convention, we consider the graviton momenta incoming as well as k_{1}, instead of the out coming k_{2} momentum as pictured in Fig. 1. In order to simplify the notation, we introduce the tensor components

 \displaystyle A^{\mu_{1}\nu_{1}\mu\nu} \displaystyle\equiv\delta^{\mu_{1}\nu_{1}}\delta^{\mu\nu}-2\delta^{\mu(\mu_{1}% }\delta^{\nu_{1})\nu} \displaystyle B^{\mu_{1}\nu_{1}\mu\nu} \displaystyle\equiv\delta^{\mu_{1}\nu_{1}}\delta^{\mu\nu}-\delta^{\mu(\mu_{1}}% \delta^{\nu_{1})\nu} \displaystyle C^{\mu_{1}\nu_{1}\mu_{2}\nu_{2}\mu\nu} \displaystyle\equiv\delta^{\mu(\mu_{1}}\delta^{\nu_{1})(\mu_{2}}\delta^{\nu_{2% })\nu}+\delta^{\mu(\mu_{2}}\delta^{\nu_{2})(\mu_{1}}\delta^{\nu_{1})\nu} \displaystyle\tilde{C}^{\mu_{1}\nu_{1}\mu_{2}\nu_{2}\mu\nu} \displaystyle\equiv\delta^{\mu(\mu_{1}}\delta^{\nu_{1})(\mu_{2}}\delta^{\nu_{2% })\nu} \displaystyle D^{\mu_{1}\nu_{1}\mu_{2}\nu_{2}\mu\nu} \displaystyle\equiv\delta^{\mu_{1}\nu_{1}}\delta^{\mu(\mu_{2}}\delta^{\nu_{2})% \nu}+\delta^{\mu_{2}\nu_{2}}\delta^{\mu(\mu_{1}}\delta^{\nu_{1})\nu} \displaystyle E^{\mu_{1}\nu_{1}\mu_{2}\nu_{2}\mu\nu} \displaystyle\equiv\delta^{\mu_{1}\nu_{1}}B^{\mu_{2}\nu_{2}\mu\nu}+C^{\mu_{1}% \nu_{1}\mu_{2}\nu_{2}\mu\nu}, \displaystyle F^{\alpha_{1}\alpha_{2}\mu\nu} \displaystyle\equiv\delta^{\alpha_{1}[\mu}\delta^{\nu]\alpha_{2}} \displaystyle\tilde{F}^{\alpha_{1}\alpha_{2}\mu\nu} \displaystyle\equiv\delta^{\alpha_{1}(\nu}\delta^{\mu)\alpha_{2}} \displaystyle\tilde{F}^{\alpha_{1}\alpha_{2}}_{\mu\nu} \displaystyle\equiv\delta^{(\alpha_{1}}_{\nu}\delta_{\mu}^{\alpha_{2})} \displaystyle G^{\mu_{1}\nu_{1}\alpha_{1}\alpha_{2}\mu\nu} \displaystyle\equiv\delta^{\mu[\nu}\delta^{\alpha_{2}](\mu_{1}}\delta^{\nu_{1}% )\alpha_{1}}+\delta^{\alpha_{1}[\alpha_{2}}\delta^{\nu](\mu_{1}}\delta^{\nu_{1% })\mu} \displaystyle H^{\mu_{1}\nu_{1}\mu_{2}\nu_{1}\alpha_{1}\alpha_{2}\mu\nu} \displaystyle\equiv A^{\mu_{1}\nu_{1}\mu\alpha_{1}}\tilde{F}^{\mu_{2}\nu_{2}% \nu\alpha_{2}}-A^{\mu_{2}\nu_{2}\mu\alpha_{1}}\tilde{F}^{\mu_{1}\nu_{1}\nu% \alpha_{2}} \displaystyle I^{\mu_{1}\nu_{1}\mu_{2}\nu_{2}\alpha_{1}\alpha_{1}\mu\nu} \displaystyle\equiv\delta^{\mu_{1}\nu_{1}}D^{\mu\alpha_{1}\nu\alpha_{2}\mu_{2}% \nu_{2}}-\mbox{\small$\displaystyle\frac{1}{2}$}\delta^{\alpha_{1}\mu}\delta^{% \alpha_{2}\nu}A^{\mu_{1}\nu_{1}\mu_{2}\nu_{2}} (D.1)

where we indicate with the circle brackets the symmetrization of the indices and the square brackets the anti-symmetrization of the indices, as follows

 \displaystyle\delta^{\mu(\mu_{1}}\delta^{\nu_{1})\nu} \displaystyle\equiv\mbox{\small$\displaystyle\frac{1}{2}$}\bigg{(}\delta^{\mu% \mu_{1}}\delta^{\nu_{1}\nu}+\delta^{\mu\nu_{1}}\delta^{\mu_{1}\nu}\bigg{)} \displaystyle\delta^{\mu[\mu_{1}}\delta^{\nu_{1}]\nu} \displaystyle\equiv\mbox{\small$\displaystyle\frac{1}{2}$}\bigg{(}\delta^{\mu% \mu_{1}}\delta^{\nu_{1}\nu}-\delta^{\mu\nu_{1}}\delta^{\mu_{1}\nu}\bigg{)} (D.2)

In the scalar sectors we obtain

 \displaystyle V^{\mu_{1}\nu_{1}}_{T\phi\phi}(k_{1},k_{2}) \displaystyle=\mbox{\small$\displaystyle\frac{1}{2}$}A^{\mu_{1}\nu_{1}\mu\nu}% \,k_{1\nu}\,k_{2\mu}+\chi\,B^{\mu_{1}\nu_{1}\mu\nu}\,(k_{1}-k_{2})_{\mu}\,(k_{% 1}-k_{2})_{\nu} (D.3) \displaystyle V^{\mu_{1}\nu_{1}\mu_{2}\nu_{2}}_{TT\phi\phi}(p_{2},k_{1},k_{2}) \displaystyle=\left(\mbox{\small$\displaystyle\frac{1}{4}$}A^{\mu_{1}\nu_{1}% \mu_{2}\nu_{2}}\delta^{\mu\nu}+C^{\mu_{1}\nu_{1}\mu_{2}\mu_{2}\mu\nu}-\mbox{% \small$\displaystyle\frac{1}{2}$}D^{\mu_{1}\nu_{1}\mu_{2}\nu_{2}\mu\nu}\right)% \,k_{1\nu}\,k_{2\mu} \displaystyle+\mbox{\small$\displaystyle\frac{\chi}{2}$}\bigg{[}\mbox{\small$% \displaystyle\frac{1}{2}$}\,\Big{(}E^{\mu_{1}\nu_{1}\mu_{2}\nu_{2}\mu\nu}-D^{% \mu_{2}\nu_{2}\mu\nu\mu_{1}\nu_{1}}\Big{)}\,p_{2\mu}p_{2\nu}+\mbox{\small$% \displaystyle\frac{1}{2}$}\,\Big{(}E^{\mu_{1}\nu_{1}\mu_{2}\nu_{2}\mu\nu}-D^{% \mu_{1}\nu_{1}\mu\nu\mu_{2}\nu_{2}}\Big{)}\,p_{2\mu}(k_{2}-k_{1})_{\nu} \displaystyle         +\Big{(}C^{\mu_{1}\nu_{1}\mu_{2}\nu_{2}\mu\nu}-D^{\mu_{1% }\nu_{1}\mu\nu\mu_{2}\nu_{2}}\Big{)}(k_{2}-k_{1})_{\mu}(k_{2}-k_{1})_{\nu}% \bigg{]}. (D.4)

In the fermion sector

 \displaystyle V^{\mu_{1}\nu_{1}}_{T\bar{\psi}\psi}(k_{1},k_{2}) \displaystyle=\mbox{\small$\displaystyle\frac{1}{4}$}\,B^{\mu_{1}\nu_{1}\mu\nu% }\,\gamma_{\nu}\,(k_{1}+k_{2})_{\mu} (D.5) \displaystyle V^{\mu_{1}\nu_{1}\mu_{2}\nu_{2}}_{TT\bar{\psi}\psi}(p_{2},k_{1},% k_{2}) \displaystyle=\mbox{\small$\displaystyle\frac{1}{8}$}\bigg{[}\delta^{\mu\nu}A^% {\mu_{1}\nu_{1}\mu_{2}\nu_{2}}-D^{\mu_{1}\nu_{1}\mu_{2}\nu_{2}\mu\nu}+C^{\mu_{% 1}\nu_{1}\mu_{2}\nu_{2}\mu\nu}+\tilde{C}^{\mu_{2}\nu_{2}\mu_{1}\nu_{1}\mu\nu}% \bigg{]}\,\gamma_{\nu}\,(k_{1}+k_{2})_{\mu} \displaystyle         +\mbox{\small$\displaystyle\frac{1}{32}$}\tilde{C}^{\mu_% {1}\nu_{1}\mu_{2}\nu_{2}\nu\mu}\,p_{2}^{\sigma}\,\left\{\gamma_{\nu},\left[% \gamma_{\mu},\gamma_{\sigma}\right]\,\right\} (D.6)

where we notice that the spin connection contribute to the two gravitons and two fermions vertex. However one can prove that this term does not contribute to the bubble diagrams.

In the gauge sector we split the contribution of the Maxwell action with respect to the gauge fixing contribution. We labelled the first type of contribution to the vertices with a subscript M, instead of the second with GF. Then in this case we obtain for the pure gauge term

 \displaystyle V^{\mu_{1}\nu_{1}\alpha_{1}\alpha_{2}}_{TAA,\,M}(k_{1},k_{2}) \displaystyle=\bigg{(}\delta^{\mu_{1}\nu_{1}}F^{\alpha_{1}\mu\nu\alpha_{2}}+2% \,G^{\mu_{1}\nu_{1}\alpha_{1}\alpha_{2}\mu\nu}\bigg{)}\,k_{1\mu}\,k_{2\nu} (D.7) \displaystyle V^{\mu_{1}\nu_{1}\mu_{2}\nu_{2}\alpha_{1}\alpha_{2}}_{TTAA,\,M}(% k_{1},k_{2}) \displaystyle=\bigg{[}-\mbox{\small$\displaystyle\frac{1}{2}$}A^{\mu_{1}\nu_{1% }\mu_{2}\nu_{2}}F^{\mu\alpha_{1}\nu\alpha_{2}}+\delta^{\mu_{2}\nu_{2}}\,G^{\mu% _{1}\nu_{1}\alpha_{1}\alpha_{2}\mu\nu}+\delta^{\mu_{1}\nu_{1}}\,G^{\mu_{2}\nu_% {2}\alpha_{1}\alpha_{2}\mu\nu} \displaystyle-\big{(}\delta^{\alpha_{1}\alpha_{2}}C^{\mu_{1}\nu_{1}\mu_{2}\nu_% {2}\mu\nu}+\delta^{\mu\nu}C^{\mu_{1}\nu_{1}\mu_{2}\nu_{2}\alpha_{1}\alpha_{2}}% -\delta^{\alpha_{1}\nu}C^{\mu_{1}\nu_{1}\mu_{2}\nu_{2}\alpha_{2}\mu}-\delta^{% \alpha_{2}\mu}C^{\mu_{1}\nu_{1}\mu_{2}\nu_{2}\alpha_{1}\nu}\big{)} \displaystyle-\big{(}\tilde{F}^{\mu\nu\mu_{1}\nu_{1}}\tilde{F}^{\mu_{2}\nu_{2}% \alpha_{1}\alpha_{2}}+\tilde{F}^{\mu\nu\mu_{2}\nu_{2}}\tilde{F}^{\mu_{1}\nu_{1% }\alpha_{1}\alpha_{2}}-\tilde{F}^{\mu\alpha_{2}\mu_{1}\nu_{1}}\tilde{F}^{\mu_{% 2}\nu_{2}\alpha_{1}\nu}-\tilde{F}^{\mu\alpha_{2}\mu_{2}\nu_{2}}\tilde{F}^{\mu_% {1}\nu_{1}\nu\alpha_{1}}\big{)}\,\bigg{]}\,k_{1\mu}\,k_{2\nu} (D.8)

and for the gauge fixing term

 \displaystyle V^{\mu_{1}\nu_{1}\alpha_{1}\alpha_{2}}_{TAA,\,GF}(k_{1},k_{2}) \displaystyle=-\mbox{\small$\displaystyle\frac{1}{2\xi}$}\bigg{[}-\delta^{\mu_% {1}\nu_{1}}\delta^{\alpha_{1}\mu}\delta^{\alpha_{2}\nu}\,k_{1\mu}\,k_{2\nu}+% \big{(}\delta^{\mu_{1}\nu_{1}}\tilde{F}^{\alpha_{1}\alpha_{2}\mu\nu}-2\tilde{C% }^{\mu_{1}\nu_{1}\mu\nu\alpha_{1}\alpha_{2}}\big{)}\,k_{2\mu}\,k_{2\nu} \displaystyle                 +\big{(}\delta^{\mu_{1}\nu_{1}}\tilde{F}^{\alpha% _{1}\alpha_{2}\mu\nu}-2\tilde{C}^{\mu_{1}\nu_{1}\mu\nu\alpha_{2}\alpha_{1}}% \big{)}\,k_{1\mu}\,k_{1\nu}\,\bigg{]}. (D.9)
 \displaystyle V^{\mu_{1}\nu_{1}\mu_{2}\nu_{2}\alpha_{1}\alpha_{2}}_{TTAA,\,GF}% (p_{2},k_{1},k_{2}) \displaystyle=-\mbox{\small$\displaystyle\frac{1}{2\xi}$}\bigg{[}I^{\mu_{1}\nu% _{1}\mu_{2}\nu_{2}\alpha_{1}\alpha_{2}\mu\nu}\,k_{1\mu}\,k_{2\nu}+H^{\mu_{2}% \nu_{2}\mu_{1}\nu_{1}\alpha_{2}\alpha_{1}\mu\nu}\,p_{2\mu}k_{1\nu} \displaystyle+H^{\mu_{1}\nu_{1}\mu_{2}\nu_{2}\alpha_{1}\alpha_{2}\mu\nu}\,p_{2% \mu}k_{2\nu}-\big{(}I^{\mu_{1}\nu_{1}\mu_{2}\nu_{2}\alpha_{1}\alpha_{2}\mu\nu}% +A^{\mu_{2}\nu_{2}\alpha_{2}\nu}\tilde{F}^{\mu_{1}\nu_{1}\mu\alpha_{1}}-2% \delta^{\alpha_{2}\nu}C^{\mu_{1}\nu_{1}\mu_{2}\nu_{2}\mu\alpha_{1}}\big{)}\,k_% {2\mu}\,k_{2\nu} \displaystyle-\big{(}I^{\mu_{1}\nu_{1}\mu_{2}\nu_{2}\alpha_{2}\alpha_{1}\mu\nu% }+A^{\mu_{2}\nu_{2}\alpha_{1}\nu}\tilde{F}^{\mu_{1}\nu_{1}\mu\alpha_{2}}-2% \delta^{\alpha_{1}\nu}C^{\mu_{1}\nu_{1}\mu_{2}\nu_{2}\mu\alpha_{2}}\big{)}\,k_% {1\mu}\,k_{1\nu}+\big{(}4\,\tilde{F}^{\mu_{1}\nu_{1}\nu(\alpha_{1}}\tilde{F}^{% \alpha_{2})\mu\mu_{2}\nu_{2}} \displaystyle-2\delta^{\mu_{1}\nu_{1}}\tilde{C}^{\alpha_{1}\alpha_{2}\mu_{2}% \nu_{2}\nu\mu}-2\delta^{\mu_{2}\nu_{2}}\tilde{C}^{\alpha_{1}\alpha_{2}\mu_{1}% \nu_{1}\mu\nu}+\delta^{\mu_{1}\nu_{1}}\delta^{\mu_{2}\nu_{2}}\tilde{F}^{\mu\nu% \alpha_{1}\alpha_{2}}\big{)}\,p_{2\mu}\,(p_{2}-k_{2}+k_{1})_{\nu}\bigg{]} (D.10)

For the ghost sector the vertices are

 \displaystyle V^{\mu_{1}\nu_{1}}_{T\bar{c}c}(k_{1},k_{2}) \displaystyle=\mbox{\small$\displaystyle\frac{1}{2}$}\,A^{\mu_{1}\nu_{1}\mu\nu% }\,k_{1\mu}\,k_{2\nu} \displaystyle V^{\mu_{1}\nu_{1}\mu_{2}\nu_{2}}_{TT\bar{c}c}(k_{1},k_{2}) \displaystyle=\bigg{(}\mbox{\small$\displaystyle\frac{1}{4}$}\delta^{\mu_{2}% \nu_{2}}\,A^{\mu_{1}\nu_{1}\mu\nu}-\mbox{\small$\displaystyle\frac{1}{2}$}\,D^% {\mu_{1}\nu_{1}\mu\nu\mu_{2}\nu_{2}}+C^{\mu_{1}\nu_{1}\mu_{2}\nu_{2}\mu\nu}% \bigg{)}\,k_{1\mu}\,k_{2\nu} (D.11)

Note that in order to find the vertices, we have to consider a i complex factor coming from the generating functional and the Fourier transformation is conventionally set in these expressions with the exponential factor \exp[-i(px-qy)] if p is an incoming momentum and q is outgoing.

Appendix E Metric variations of the counterterms

In this appendix we list the metric variations of the counterterms. In particular we give them directly in the momentum using the definition of the Fourier transform in (10.4). The metric variation are consider in the flat space-time limit and the first variation of the square of the metric, Riemann, Ricci and the scalar curvature are given as

 \displaystyle\big{[}\sqrt{-g}\big{]}^{\mu_{i}\nu_{i}} \displaystyle=\mbox{\small$\displaystyle\frac{1}{2}$}\delta^{\mu_{i}\nu_{i}} \displaystyle\big{[}R_{\mu\alpha\nu\beta}\big{]}^{\mu_{i}\nu_{i}}(p_{i}) \displaystyle=\frac{1}{2}\,\bigg{(}\delta_{\alpha}^{(\mu_{i}}\delta^{\nu_{i})}% _{\beta}\,p_{i\mu}\,p_{i\nu}+\delta_{\mu}^{(\mu_{i}}\delta^{\nu_{i})}_{\nu}\,p% _{i\alpha}\,p_{i\beta}-\delta_{\mu}^{(\mu_{i}}\delta^{\nu_{i})}_{\beta}\,p_{i% \alpha}\,p_{i\nu}-\delta_{\alpha}^{(\mu_{i}}\delta^{\nu_{i})}_{\nu}\,p_{i\mu}% \,p_{i\beta}\bigg{)} \displaystyle\big{[}R^{\mu\alpha\nu\beta}\big{]}^{\mu_{i}\nu_{i}}(p_{i}) \displaystyle=\frac{1}{2}\,\bigg{(}\delta^{\alpha(\mu_{i}}\delta^{\nu_{i})% \beta}\,p_{i}^{\mu}\,p_{i}^{\nu}+\delta^{\mu(\mu_{i}}\delta^{\nu_{i})\nu}\,p_{% i}^{\alpha}\,p_{i}^{\beta}-\delta^{\mu(\mu_{i}}\delta^{\nu_{i})\beta}\,p_{i}^{% \alpha}\,p_{i}^{\nu}-\delta^{\alpha(\mu_{i}}\delta^{\nu_{i})\nu}\,p_{i}^{\mu}% \,p_{i}^{\beta}\bigg{)} \displaystyle\big{[}R_{\mu\nu}\big{]}^{\mu_{i}\nu_{i}}(p_{i}) \displaystyle=\frac{1}{2}\,\bigg{(}\delta_{\mu}^{(\mu_{i}}\delta^{\nu_{i})}_{% \nu}\,p_{i}^{2}+\delta^{\mu_{i}\nu_{i}}\,p_{i\mu}\,p_{i\nu}-p_{i}^{(\mu_{i}}% \delta^{\nu_{i})}_{\mu}\,p_{i\nu}-p_{i}^{(\mu_{i}}\delta^{\nu_{i})}_{\nu}\,p_{% i\mu}\bigg{)} \displaystyle\big{[}R^{\mu\nu}\big{]}^{\mu_{i}\nu_{i}}(p_{i}) \displaystyle=\frac{1}{2}\,\bigg{(}\delta^{\mu(\mu_{i}}\delta^{\nu_{i})\nu}\,p% _{i}^{2}+\delta^{\mu_{i}\nu_{i}}\,p_{i}^{\mu}\,p_{i}^{\nu}-p_{i}^{(\mu_{i}}% \delta^{\nu_{i})\mu}\,p_{i}^{\nu}-p_{i}^{(\mu_{i}}\delta^{\nu_{i})\nu}\,p_{i}^% {\mu}\bigg{)} \displaystyle\big{[}R\big{]}^{\mu_{i}\nu_{i}}(p_{i}) \displaystyle=\bigg{(}\delta^{\mu_{i}\nu_{i}}\,p_{i}^{2}-p_{i}^{(\mu_{i}}p_{i}% ^{\nu_{i})}\bigg{)} \displaystyle\big{[}\square R\big{]}^{\mu_{i}\nu_{i}}(p_{i}) \displaystyle=p_{i}^{2}\,\bigg{(}p_{i}^{(\mu_{i}}p_{i}^{\nu_{i})}-\delta^{\mu_% {i}\nu_{i}}\,p_{i}^{2}\bigg{)} (E.1)

and the second variations of these object can be calculated in order to obtain

 \displaystyle\big{[}R^{\,\beta}_{\ \ \nu\rho\sigma}\big{]}^{\mu_{1}\nu_{1}\mu_% {2}\nu_{2}}(p_{1},p_{2}) \displaystyle=\Big{[}-\frac{1}{2}\,\tilde{F}^{\mu_{1}\nu_{1}\beta\epsilon}p_{1% \sigma}\big{(}\tilde{F}^{\mu_{2}\nu_{2}}_{\epsilon\nu}p_{2\rho}+\tilde{F}^{\mu% _{2}\nu_{2}}_{\epsilon\rho}p_{2\nu}-\tilde{F}^{\mu_{2}\nu_{2}}_{\nu\rho}p_{2% \epsilon}\big{)} \displaystyle-\frac{1}{2}\big{(}\tilde{C}^{\mu_{1}\nu_{1}\mu_{2}\nu_{2}\beta}_% {\hskip 39.833858pt\rho}\ p_{2\nu}-\tilde{F}^{\mu_{1}\nu_{1}\beta\epsilon}\,% \tilde{F}^{\mu_{2}\nu_{2}}_{\nu\rho}\,p_{2\epsilon}\big{)}p_{2\sigma} \displaystyle-\frac{1}{4}\big{(}\tilde{F}^{\mu_{1}\nu_{1}}_{\alpha\nu}p_{1% \sigma}+\tilde{F}^{\mu_{1}\nu_{1}}_{\alpha\sigma}p_{1\nu}-\tilde{F}^{\mu_{1}% \nu_{1}}_{\sigma\nu}p_{1\alpha}\big{)}\big{(}\tilde{F}^{\mu_{2}\nu_{2}\beta% \alpha}\,p_{2\rho}+\tilde{F}^{\mu_{2}\nu_{2}\beta}_{\qquad\rho}\,p_{2}^{\alpha% }-\tilde{F}^{\mu_{2}\nu_{2}\alpha}_{\qquad\rho}\,p_{2}^{\beta}\big{)}\Big{]}-(% \sigma\leftrightarrow\rho) \displaystyle\big{[}R_{\mu\nu\rho\sigma}\big{]}^{\mu_{1}\nu_{1}\mu_{2}\nu_{2}}% (p_{1},p_{2}) \displaystyle=\delta^{(\mu_{1}}_{\mu}\delta^{\nu_{1})}_{\beta}\big{[}R^{\,% \beta}_{\ \ \nu\rho\sigma}\big{]}^{\mu_{2}\nu_{2}}(p_{2})+\delta_{\mu\beta}% \big{[}R^{\,\beta}_{\ \ \nu\rho\sigma}\big{]}^{\mu_{1}\nu_{1}\mu_{2}\nu_{2}}(p% _{1},p_{2})
 \displaystyle\big{[}R^{\mu\nu\rho\sigma}\big{]}^{\mu_{1}\nu_{1}\mu_{2}\nu_{2}}% (p_{1},p_{2}) \displaystyle=\delta^{\alpha\nu}\delta^{\beta\rho}\delta^{\sigma\gamma}\big{[}% R^{\,\mu}_{\ \ \alpha\beta\gamma}\big{]}^{\mu_{1}\nu_{1}\mu_{2}\nu_{2}}(p_{1},% p_{2}) \displaystyle-\big{(}\delta^{\alpha(\mu_{1}}\delta^{\nu_{1})\nu}\delta^{\beta% \rho}\delta^{\sigma\gamma}+\delta^{\alpha\nu}\delta^{\beta(\mu_{1}}\delta^{\nu% _{1})\rho}\delta^{\sigma\gamma}+\delta^{\alpha\nu}\delta^{\beta\rho}\delta^{% \sigma(\mu_{1}}\delta^{\nu_{1})\gamma}\big{)}\big{[}R^{\,\beta}_{\ \ \nu\rho% \sigma}\big{]}^{\mu_{2}\nu_{2}}(p_{2}) (E.3)
 \displaystyle\big{[}R_{\nu\sigma}\big{]}^{\mu_{1}\nu_{1}\mu_{2}\nu_{2}}(p_{1},% p_{2}) \displaystyle=-\frac{1}{2}\tilde{F}^{\mu_{1}\nu_{1}\mu_{2}\nu_{2}}\left(p_{1% \sigma}p_{2\nu}-\mbox{\small$\displaystyle\frac{1}{2}$}p_{1\nu}p_{2\sigma}+p_{% 2\nu}p_{2\sigma}\right)-\frac{1}{4}\delta^{\mu_{2}\nu_{2}}\left(\tilde{F}^{\mu% _{1}\nu_{1}}_{\alpha\nu}\,p_{1\sigma}+\tilde{F}^{\mu_{1}\nu_{1}}_{\alpha\sigma% }\,p_{1\nu}\right)\,p_{2}^{\alpha} \displaystyle+\frac{1}{2}\big{(}\tilde{C}^{\mu_{1}\nu_{1}\mu_{2}\nu_{2}% \epsilon}_{\hskip 39.833858pt\nu}\,p_{2\sigma}+\tilde{C}^{\mu_{1}\nu_{1}\mu_{2% }\nu_{2}\epsilon}_{\hskip 39.833858pt\sigma}\,p_{2\nu}\big{)}\,(p_{1}+p_{2})_{% \epsilon}+\frac{1}{2}F^{\mu_{2}\nu_{2}}_{\alpha\sigma}\tilde{F}^{\mu_{1}\nu_{1% }}_{\beta\nu}\,p_{1}^{\alpha}\,p_{2}^{\beta} \displaystyle-\frac{1}{2}\tilde{F}^{\mu_{2}\nu_{2}}_{\nu\sigma}\,\tilde{F}^{% \mu_{1}\nu_{1}\alpha\beta}(p_{1}+p_{2})_{\alpha}\,p_{2\beta}-\frac{1}{2}\left(% \tilde{C}^{\mu_{1}\nu_{1}\mu_{2}\nu_{2}}_{\hskip 35.565945pt\nu\sigma}-\frac{1% }{2}\delta^{\mu_{2}\nu_{2}}\,\tilde{F}^{\mu_{1}\nu_{1}}_{\nu\sigma}\right)\,p_% {1}\cdot p_{2}
 \displaystyle\big{[}R^{\nu\sigma}\big{]}^{\mu_{1}\nu_{1}\mu_{2}\nu_{2}}(p_{1},% p_{2}) \displaystyle=\delta^{\nu\alpha}\delta^{\sigma\beta}\big{[}R_{\alpha\beta}\big% {]}^{\mu_{1}\nu_{1}\mu_{2}\nu_{2}}(p_{1},p_{2})-\big{(}\delta^{\nu(\mu_{1}}% \delta^{\nu_{1})\alpha}\delta^{\sigma\beta}+\delta^{\nu\alpha}\delta^{\sigma(% \mu_{1}}\delta^{\nu_{1})\beta}\big{)}\big{[}R_{\alpha\beta}\big{]}^{\mu_{2}\nu% _{2}}(p_{2}) (E.5) \displaystyle\big{[}R\big{]}^{\mu_{1}\nu_{1}\mu_{2}\nu_{2}}(p_{1},p_{2}) \displaystyle=-\left(p_{2}^{2}+\mbox{\small$\displaystyle\frac{1}{4}$}p_{1}% \cdot p_{2}\right)\,\tilde{F}^{\mu_{1}\nu_{1}\mu_{2}\nu_{2}}+\frac{1}{4}\,A^{% \mu_{1}\nu_{1}\mu_{2}\nu_{2}}\,p_{1}\cdot p_{2} \displaystyle+\tilde{C}^{\mu_{1}\nu_{1}\mu_{2}\nu_{2}\alpha\beta}\,(p_{1\alpha% }+2p_{2\alpha})p_{2\beta}-\delta^{\mu_{2}\nu_{2}}\tilde{F}^{\mu_{1}\nu_{1}% \alpha\beta}\,(p_{1\alpha}+p_{2\alpha})p_{2\beta}+\mbox{\small$\displaystyle% \frac{1}{2}$}\tilde{C}^{\mu_{2}\nu_{2}\mu_{1}\nu_{1}\alpha\beta}\,p_{1\alpha}p% _{2\beta} (E.6)
 \displaystyle\big{[}\square R\big{]}^{\mu_{1}\nu_{1}\mu_{2}\nu_{2}}(p_{1},p_{2}) \displaystyle=\tilde{F}^{\mu_{1}\nu_{1}\mu_{2}\nu_{2}}\,\bigg{[}p_{2}^{2}(p_{1% }+p_{2})^{2}+\mbox{\small$\displaystyle\frac{3}{2}$}(p_{2}^{2}+p_{1}\cdot p_{2% })\bigg{]}+\mbox{\small$\displaystyle\frac{1}{2}$}\delta^{\mu_{1}\nu_{1}}% \tilde{F}^{\mu_{2}\nu_{2}\alpha\beta}(p_{1}\cdot p_{2})\,p_{2\alpha}p_{2\beta} \displaystyle-\mbox{\small$\displaystyle\frac{1}{2}$}\delta^{\mu_{1}\nu_{1}}% \delta^{\mu_{2}\nu_{2}}(p_{1}\cdot p_{2})\bigg{[}(p_{1}+p_{2})^{2}-p_{1}\cdot p% _{2}\bigg{]}+\delta^{\mu_{2}\nu_{2}}F^{\mu_{1}\nu_{1}\alpha\beta}p_{2\alpha}(p% _{1}+p_{2})_{\beta}\bigg{[}(p_{1}+p_{2})^{2}+p_{2}^{2}\bigg{]} \displaystyle-\tilde{F}^{\mu_{2}\nu_{2}\alpha\beta}p_{2\alpha}p_{2\beta}\,% \tilde{F}^{\mu_{1}\nu_{1}\gamma\delta}p_{2\gamma}(p_{1}+p_{2})_{\delta}-(p_{1}% +p_{2})^{2}\,\tilde{C}^{\mu_{1}\nu_{1}\mu_{2}\nu_{2}\alpha\beta}\bigg{[}2p_{2% \alpha}p_{2\beta}+p_{1\alpha}p_{2\beta}+\mbox{\small$\displaystyle\frac{1}{2}$% }p_{2\alpha}p_{1\beta}\bigg{]} (E.7)

remembering that the order of indices of variation is important, because these second variation are not symmetrized.

Using these relations it is possible to find the third variation of the counterterms. For instance the Weyl tensor counterterm is expressed as

 \displaystyle\big{[}\sqrt{-g}\,C^{2}\big{]}^{\mu_{1}\nu_{1}\mu_{2}\nu_{2}\mu_{% 3}\nu_{3}}(p_{1},p_{2},p_{3}) \displaystyle=\Bigg{\{}[\sqrt{-g}]^{\mu_{1}\nu_{1}}\bigg{(}[R_{abcd}]^{\mu_{2}% \nu_{2}}(p_{2})[R^{abcd}]^{\mu_{3}\nu_{3}}(p_{3})-\mbox{\small$\displaystyle% \frac{4}{d-2}$}[R_{ab}]^{\mu_{2}\nu_{2}}(p_{2})[R^{ab}]^{\mu_{3}\nu_{3}}(p_{3}) \displaystyle+\mbox{\small$\displaystyle\frac{2}{(d-2)(d-1)}$}[R]^{\mu_{2}\nu_% {2}}(p_{2})[R]^{\mu_{3}\nu_{3}}(p_{3})\bigg{)}+\bigg{(}[R_{abcd}]^{\mu_{1}\nu_% {1}\mu_{2}\nu_{2}}(p_{1},p_{2})[R^{abcd}]^{\mu_{3}\nu_{3}}(p_{3}) \displaystyle+[R_{abcd}]^{\mu_{2}\nu_{2}}(p_{2})[R^{abcd}]^{\mu_{1}\nu_{1}\mu_% {3}\nu_{3}}(p_{1},p_{3})-\mbox{\small$\displaystyle\frac{4}{d-2}$}[R_{ab}]^{% \mu_{1}\nu_{1}\mu_{2}\nu_{2}}(p_{1},p_{2})[R^{ab}]^{\mu_{3}\nu_{3}}(p_{3}) \displaystyle-\mbox{\small$\displaystyle\frac{4}{d-2}$}[R_{ab}]^{\mu_{2}\nu_{2% }}(p_{2})[R^{ab}]^{\mu_{1}\nu_{1}\mu_{3}\nu_{3}}(p_{1},p_{3})+\mbox{\small$% \displaystyle\frac{2}{(d-2)(d-1)}$}[R]^{\mu_{1}\nu_{1}\mu_{2}\nu_{2}}(p_{1},p_% {2})[R]^{\mu_{3}\nu_{3}}(p_{3}) \displaystyle+\mbox{\small$\displaystyle\frac{2}{(d-2)(d-1)}$}[R]^{\mu_{2}\nu_% {2}}(p_{2})[R]^{\mu_{1}\nu_{1}\mu_{3}\nu_{3}}(p_{1},p_{3})\bigg{)}\Bigg{\}}+% \text{permutations} (E.8)

where “permutation” indicate all the possible permutations of the indices (\mu_{i},\nu_{i}). In a same way the Euler density counterterm is given as

 \displaystyle\big{[}\sqrt{-g}\,E\big{]}^{\mu_{1}\nu_{1}\mu_{2}\nu_{2}\mu_{3}% \nu_{3}}(p_{1},p_{2},p_{3}) \displaystyle=\Bigg{\{}[\sqrt{-g}]^{\mu_{1}\nu_{1}}\bigg{(}[R_{abcd}]^{\mu_{2}% \nu_{2}}(p_{2})[R^{abcd}]^{\mu_{3}\nu_{3}}(p_{3})-4[R_{ab}]^{\mu_{2}\nu_{2}}(p% _{2})[R^{ab}]^{\mu_{3}\nu_{3}}(p_{3}) \displaystyle+[R]^{\mu_{2}\nu_{2}}(p_{2})[R]^{\mu_{3}\nu_{3}}(p_{3})\bigg{)}+% \bigg{(}[R_{abcd}]^{\mu_{1}\nu_{1}\mu_{2}\nu_{2}}(p_{1},p_{2})[R^{abcd}]^{\mu_% {3}\nu_{3}}(p_{3})+[R_{abcd}]^{\mu_{2}\nu_{2}}(p_{2})[R^{abcd}]^{\mu_{1}\nu_{1% }\mu_{3}\nu_{3}}(p_{1},p_{3}) \displaystyle-4[R_{ab}]^{\mu_{1}\nu_{1}\mu_{2}\nu_{2}}(p_{1},p_{2})[R^{ab}]^{% \mu_{3}\nu_{3}}(p_{3})-4[R_{ab}]^{\mu_{2}\nu_{2}}(p_{2})[R^{ab}]^{\mu_{1}\nu_{% 1}\mu_{3}\nu_{3}}(p_{1},p_{3})+[R]^{\mu_{1}\nu_{1}\mu_{2}\nu_{2}}(p_{1},p_{2})% [R]^{\mu_{3}\nu_{3}}(p_{3}) \displaystyle+[R]^{\mu_{2}\nu_{2}}(p_{2})[R]^{\mu_{1}\nu_{1}\mu_{3}\nu_{3}}(p_% {1},p_{3})\bigg{)}\Bigg{\}}+\text{permutations}. (E.9)

Appendix F Form Factors in d=5

 \displaystyle A_{2}^{D=5}(p_{1},p_{2},p_{3}) \displaystyle=\frac{\pi^{4}(n_{S}-4\,n_{F})}{1680(p_{1}+p_{2}+p_{3})^{6}}\Big{% [}-24p_{3}^{5}\big{(}8(p_{1}+p_{2})^{2}-p_{1}p_{2}\big{)}-24p_{3}^{4}(p_{1}+p_% {2})\big{(}13(p_{1}+p_{2})-6p_{1}p_{2}\big{)} \displaystyle-8p_{3}^{3}\left(42(p_{1}+p_{2})^{4}-p_{1}p_{2}\left(33(p_{1}+p_{% 2})^{2}+8p_{1}p_{2}\right)\right) \displaystyle-5(6p_{3}+p_{1}+p_{2})(p_{1}+p_{2})^{2}\big{(}3(p_{1}+p_{2})^{4}-% p_{1}p_{2}\left(3(p_{1}+p_{2})^{2}+p_{1}p_{2}\right)\big{)} \displaystyle-3p_{3}^{2}(p_{1}+p_{2})\left(77(p_{1}+p_{2})^{4}-p_{1}p_{2}\left% (63(p_{1}+p_{2})^{2}+43p_{1}p_{2}\right)\right)-72p_{3}^{6}(p_{1}+p_{2})-12p_{% 3}^{7}\Big{]} \displaystyle+\frac{\pi^{4}\,n_{F}}{40(p_{1}+p_{2}+p_{3})^{5}}\Big{[}-3p_{1}^{% 5}(p_{1}+5p_{2}+5p_{3})-4p_{1}^{4}\left(8(p_{2}+p_{3})^{2}-p_{2}p_{3}\right) \displaystyle-20p_{1}^{3}(p_{2}+p_{3})\left(2(p_{2}+p_{3})^{2}-p_{2}p_{3}% \right)-4p_{1}^{2}\left(8(p_{2}+p_{3})^{4}-p_{2}p_{3}\left(7(p_{2}+p_{3})^{2}+% 2p_{2}p_{3}\right)\right) \displaystyle-(p_{2}+p_{3})(5p_{1}+p_{2}+p_{3})\left(3(p_{2}+p_{3})^{4}-3p_{2}% p_{3}(p_{2}+p_{3})^{2}-p_{2}^{2}p_{3}^{2}\right)\Big{]}] (F.1)
 \displaystyle A_{3}^{D=5}(p_{1},p_{2},p_{3}) \displaystyle=\frac{\pi^{4}(n_{S}-4n_{F})p_{3}^{2}}{20160(p_{1}+p_{2}+p_{3})^{% 5}}\Big{[}-8p_{3}^{2}\left(-13p_{1}^{2}p_{2}^{2}+87(p_{1}+p_{2})^{4}-63p_{1}p_% {2}(p_{1}+p_{2})^{2}\right) \displaystyle-100p_{3}(p_{1}+p_{2})\left(-p_{1}^{2}p_{2}^{2}+3(p_{1}+p_{2})^{4% }-3p_{1}p_{2}(p_{1}+p_{2})^{2}\right)-20(p_{1}+p_{2})^{2}\big{(}-p_{1}^{2}p_{2% }^{2}+3(p_{1}+p_{2})^{4} \displaystyle-3p_{1}p_{2}(p_{1}+p_{2})^{2}\big{)}-405p_{3}^{5}(p_{1}+p_{2})-3p% _{3}^{4}\left(281(p_{1}+p_{2})^{2}-22p_{1}p_{2}\right)-15p_{3}^{3}(p_{1}+p_{2}% )\big{(}65(p_{1}+p_{2})^{2} \displaystyle-22p_{1}p_{2}\big{)}-81p_{3}^{6}\Big{]}+\frac{\pi^{4}\,n_{F}\,p_{% 3}^{2}}{240(p_{1}+p_{2}+p_{3})^{4}}\Big{[}-8p_{3}\left(-p_{1}^{2}p_{2}^{2}+3(p% _{1}+p_{2})^{4}-3p_{1}p_{2}(p_{1}+p_{2})^{2}\right) \displaystyle-2(p_{1}+p_{2})\left(-p_{1}^{2}p_{2}^{2}+3(p_{1}+p_{2})^{4}-3p_{1% }p_{2}(p_{1}+p_{2})^{2}\right)-36p_{3}^{4}(p_{1}+p_{2})-3p_{3}^{3}\left(19(p_{% 1}+p_{2})^{2}-2p_{1}p_{2}\right) \displaystyle-24p_{3}^{2}(p_{1}+p_{2})\left(2(p_{1}+p_{2})^{2}-p_{1}p_{2}% \right)-9p_{3}^{5}\Big{]}+\frac{\pi^{4}(n_{S}+8n_{F})}{576(p_{1}+p_{2}+p_{3})^% {3}}\Big{[}2p_{1}^{2}p_{2}^{2}p_{3}^{2} \displaystyle+3p_{1}p_{2}p_{3}(p_{1}+p_{2}+p_{3})\left((p_{1}+p_{2}+p_{3})^{2}% +p_{1}p_{2}+p_{1}p_{3}+p_{2}p_{3}\right)+(p_{1}p_{2}+p_{1}p_{3}+p_{2}p_{3})^{2} \displaystyle+3(p_{1}+p_{2}+p_{3})^{2}\left((p_{1}+p_{2}+p_{3})^{4}-3(p_{1}p_{% 2}+p_{1}p_{3}+p_{2}p_{3})(p_{1}+p_{2}+p_{3})^{2}\right)\Big{]} (F.2)
 \displaystyle A_{4}^{D=5}(p_{1},p_{2},p_{3}) \displaystyle=\frac{\pi^{4}(n_{S}-4n_{F})p_{3}^{2}}{10080(p_{1}+p_{2}+p_{3})^{% 5}}\Big{[}45p_{3}^{8}+225p_{3}^{7}(p_{1}+p_{2})+15p_{3}^{6}\big{(}29(p_{1}+p_{% 2})^{2}+2p_{1}p_{2}\big{)} \displaystyle+75p_{3}^{5}(p_{1}+p_{2})\big{(}5(p_{1}+p_{2})^{2}+2p_{1}p_{2}% \big{)}+8p_{3}^{4}\big{(}75(p_{1}+p_{2})^{2}-23p_{1}p_{2}\big{)}p_{1}p_{2} \displaystyle-5p_{3}^{3}(p_{1}+p_{2})\big{(}75(p_{1}+p_{2})^{4}-255(p_{1}+p_{2% })^{4}-255(p_{1}+p_{2})^{2}p_{1}p_{2}+79p_{1}^{2}p_{2}^{2}\big{)} \displaystyle-p_{3}^{2}\big{(}435(p_{1}+p_{2})^{6}-1335(p_{1}+p_{2})^{4}p_{1}p% _{2}+343(p_{1}+p_{2})^{2}p_{1}^{2}p_{2}^{2}-96p_{1}^{3}p_{2}^{3}\big{)} \displaystyle-3(p_{1}+p_{2})(5p_{3}+(p_{1}+p_{2}))(15(p_{1}+p_{2})^{6}-45(p_{1% }+p_{2})^{4}p_{1}p_{2}+11(p_{1}+p_{2})^{2}p_{1}^{2}p_{2}^{2}-4p_{1}^{3}p_{2}^{% 3})\Big{]} \displaystyle+\frac{\pi^{4}\,n_{F}}{240(p_{1}+p_{2}+p_{3})^{4}}\Big{[}9p_{3}^{% 7}+36\,p_{3}^{6}(p_{1}+p_{2})+3p_{3}^{5}\big{(}17(p_{1}+p_{2})^{2}+2p_{1}p_{2}% \big{)} \displaystyle+24p_{3}^{4}(p_{1}+p_{2})\big{(}(p_{1}+p_{2})^{2}+p_{1}p_{2}\big{% )}-8p_{3}^{3}\big{(}3(p_{1}+p_{2})^{4}-12(p_{1}+p_{2})^{2}p_{1}p_{2}+5p_{1}^{2% }p_{2}^{2}\big{)} \displaystyle-p_{3}^{2}(p_{1}+p_{2})\big{(}51(p_{1}+p_{2})^{4}-159(p_{1}+p_{2}% )^{2}p_{1}p_{2}+55p_{1}^{2}p_{2}^{2}\big{)} \displaystyle-9(p_{1}+p_{2})^{2}(4p_{3}+p_{1}+p_{2})\big{(}(p_{1}+p_{2})^{4}-3% (p_{1}+p_{2})^{2}p_{1}p_{2}+p_{1}^{2}p_{2}^{2}\big{)}\Big{]}+\frac{\pi^{4}(n_{% s}+8n_{F})}{288(p_{1}+p_{2}+p_{3})^{3}}\Big{[}2p_{1}^{2}p_{2}^{2}p_{3}^{2} \displaystyle+3(p_{1}+p_{2}+p_{3})^{2}\big{(}(p_{1}+p_{2}+p_{3})^{4}-3(p_{1}+p% _{2}+p_{3})^{2}(p_{1}p_{3}+p_{2}p_{1}+p_{3}p_{2})+(p_{1}p_{3}+p_{2}p_{1}+p_{3}% p_{2})^{2}\big{)} \displaystyle+3(p_{1}+p_{2}+p_{3})\big{(}(p_{1}+p_{2}+p_{3})^{2}+p_{1}p_{2}+p_% {2}p_{3}+p_{1}p_{3}\big{)}p_{1}p_{2}p_{3}\Big{]} (F.3)
 \displaystyle A_{5}^{D=5}(p_{1},p_{2},p_{3}) \displaystyle=\frac{\pi^{4}(n_{S}-4n_{F})}{6720(p_{1}+p_{2}+p_{3})^{4}}\Big{[}% 8p_{1}^{3}p_{2}^{3}p_{3}^{3}-4p_{1}^{2}p_{2}^{2}p_{3}^{2}(p_{1}+p_{2}+p_{3})% \big{(}(p_{1}+p_{2}+p_{3})^{2} \displaystyle-4(p_{1}p_{2}+p_{1}p_{3}+p_{2}p_{3})\big{)}+5p_{1}p_{2}p_{3}(p_{1% }+p_{2}+p_{3})^{2}\big{(}23(p_{1}+p_{2}+p_{3})^{4} \displaystyle-23(p_{1}p_{2}+p_{1}p_{3}+p_{2}p_{3})(p_{1}+p_{2}+p_{3})^{2}+4(p_% {1}p_{2}+p_{1}p_{3}+p_{2}p_{3})^{2}\big{)} \displaystyle+5(p_{1}+p_{2}+p_{3})^{3}\left((p_{1}+p_{2}+p_{3})^{2}-4(p_{1}p_{% 2}+p_{1}p_{3}+p_{2}p_{3})\right)\big{(}3(p_{1}+p_{2}+p_{3})^{4} \displaystyle-3(p_{1}p_{2}+p_{1}p_{3}+p_{2}p_{3})(p_{1}+p_{2}+p_{3})^{2}-(p_{1% }p_{2}+p_{1}p_{3}+p_{2}p_{3})^{2}\big{)}\Big{]} \displaystyle+\frac{\pi^{4}n_{F}}{480(p_{1}+p_{2}+p_{3})^{3}}\Big{[}-4p_{1}^{2% }p_{2}^{2}p_{3}^{2}\left((p_{1}+p_{2}+p_{3})^{2}-2(p_{1}p_{2}+p_{1}p_{3}+p_{2}% p_{3})\right) \displaystyle+3p_{1}p_{2}p_{3}(p_{1}+p_{2}+p_{3})\big{(}23(p_{1}+p_{2}+p_{3})^% {4}-23(p_{1}p_{2}+p_{1}p_{3}+p_{2}p_{3})(p_{1}+p_{2}+p_{3})^{2} \displaystyle+4(p_{1}p_{2}+p_{1}p_{3}+p_{2}p_{3})^{2}\big{)}+3(p_{1}+p_{2}+p_{% 3})^{2}\left((p_{1}+p_{2}+p_{3})^{2}-4(p_{1}p_{2}+p_{1}p_{3}+p_{2}p_{3})\right)\times \displaystyle\times\left(3(p_{1}+p_{2}+p_{3})^{4}-3(p_{1}p_{2}+p_{1}p_{3}+p_{2% }p_{3})(p_{1}+p_{2}+p_{3})^{2}-(p_{1}p_{2}+p_{1}p_{3}+p_{2}p_{3})^{2}\right)% \Big{]} \displaystyle+\frac{\pi^{4}(n_{S}+8n_{F})}{192}\left(p_{1}^{5}+p_{2}^{5}+p_{3}% ^{5}\right) (F.4)

Appendix G Renormalized Form Factors in d=4

We collect the explicit expressions of the renormalized form factors in d=4.
We define \sigma=s^{2}-2s(s_{1}+s_{2})+(s_{1}-s_{2})^{2}

 \displaystyle A_{1}^{Ren} \displaystyle=\pi^{2}(4n_{F}-2n_{G}-n_{S})\bigg{\{}\frac{1}{45\sigma^{5}}\bigg% {[}s^{9}-13s^{8}(s_{1}+s_{2})+2s^{7}\left(25s_{1}^{2}+77s_{1}s_{2}+25s_{2}^{2}% \right)-2s^{6}(s_{1}+s_{2})\left(41s_{1}^{2}-9s_{1}s_{2}+41s_{2}^{2}\right) \displaystyle+s^{5}\left(44s_{1}^{4}-922s_{1}^{3}s_{2}+5088s_{1}^{2}s_{2}^{2}-% 922s_{1}s_{2}^{3}+44s_{2}^{4}\right)+2s^{4}(s_{1}+s_{2})\left(22s_{1}^{4}+823s% _{1}^{3}s_{2}-3360s_{1}^{2}s_{2}^{2}+823s_{1}s_{2}^{3}+22s_{2}^{4}\right) \displaystyle-2s^{3}\left(41s_{1}^{6}+461s_{1}^{5}s_{2}+2537s_{1}^{4}s_{2}^{2}% -8598s_{1}^{3}s_{2}^{3}+2537s_{1}^{2}s_{2}^{4}+461s_{1}s_{2}^{5}+41s_{2}^{6}% \right)+2s^{2}(s_{1}-s_{2})^{2}(s_{1}+s_{2})\big{(}25s_{1}^{4}-7s_{1}^{3}s_{2} \displaystyle+2562s_{1}^{2}s_{2}^{2}-7s_{1}s_{2}^{3}+25s_{2}^{4}\big{)}-s(s_{1% }-s_{2})^{4}\left(13s_{1}^{4}-102s_{1}^{3}s_{2}-422s_{1}^{2}s_{2}^{2}-102s_{1}% s_{2}^{3}+13s_{2}^{4}\right)+(s_{1}-s_{2})^{6}(s_{1}+s_{2})\left(s_{1}^{2}-8s_% {1}s_{2}+s_{2}^{2}\right)\bigg{]} \displaystyle-\frac{4s^{2}}{15\sigma^{6}}\bigg{[}s_{2}^{5}\left(-35s^{4}-2469s% ^{2}s_{1}^{2}+2428ss_{1}^{3}+726s_{1}^{4}\right)+s_{2}^{6}\big{(}35s^{3}+135s^% {2}s_{1}+448ss_{1}^{2}-1052s_{1}^{3}\big{)}+s_{2}^{2}(s-s_{1})^{3}\big{(}s^{4}% -24s^{3}s_{1}-675s^{2}s_{1}^{2} \displaystyle-1348ss_{1}^{3}-300s_{1}^{4}\big{)}+3s_{2}^{4}\big{(}7s^{5}-45s^{% 4}s_{1}+581s^{3}s_{1}^{2}+925s^{2}s_{1}^{3}-1850ss_{1}^{4}+242s_{1}^{5}\big{)}% -s_{2}^{3}(s-s_{1})\big{(}7s^{5}-101s^{4}s_{1}-705s^{3}s_{1}^{2} \displaystyle+4151s^{2}s_{1}^{3}+1376ss_{1}^{4}-1052s_{1}^{5}\big{)}-27s_{1}^{% 2}s_{2}(s-s_{1})^{5}(s+s_{1})+s_{1}^{2}(s-s_{1})^{7}+s_{2}^{8}(7s+27s_{1})-3s_% {2}^{7}(s-2s_{1})(7s+50s_{1})-s_{2}^{9}\bigg{]}\,\bar{B}_{0}(s) \displaystyle-\frac{4s_{1}^{2}}{15\sigma^{6}}\bigg{[}-s^{9}+s^{8}(7s_{1}+27s_{% 2})-3s^{7}(s_{1}-2s_{2})(7s_{1}+50s_{2})+s^{6}\left(35s_{1}^{3}+135s_{1}^{2}s_% {2}+448s_{1}s_{2}^{2}-1052s_{2}^{3}\right)+s^{5}\big{(}-35s_{1}^{4} \displaystyle-2469s_{1}^{2}s_{2}^{2}+2428s_{1}s_{2}^{3}+726s_{2}^{4}\big{)}+3s% ^{4}\left(7s_{1}^{5}-45s_{1}^{4}s_{2}+581s_{1}^{3}s_{2}^{2}+925s_{1}^{2}s_{2}^% {3}-1850s_{1}s_{2}^{4}+242s_{2}^{5}\right)-s^{3}(s_{1}-s_{2})\big{(}7s_{1}^{5}% -101s_{1}^{4}s_{2} \displaystyle-705s_{1}^{3}s_{2}^{2}+4151s_{1}^{2}s_{2}^{3}+1376s_{1}s_{2}^{4}-% 1052s_{2}^{5}\big{)}+s^{2}(s_{1}-s_{2})^{3}\left(s_{1}^{4}-24s_{1}^{3}s_{2}-67% 5s_{1}^{2}s_{2}^{2}-1348s_{1}s_{2}^{3}-300s_{2}^{4}\right) \displaystyle-27ss_{2}^{2}(s_{1}-s_{2})^{5}(s_{1}+s_{2})+s_{2}^{2}(s_{1}-s_{2}% )^{7}\bigg{]}\bar{B}_{0}(s_{1})-\frac{4s_{2}^{2}}{15\sigma^{6}}\bigg{[}-s^{9}+% s^{8}(27s_{1}+7s_{2})+3s^{7}(2s_{1}-s_{2})(50s_{1}+7s_{2})+s^{6}\big{(}-1052s_% {1}^{3} \displaystyle+448s_{1}^{2}s_{2}+135s_{1}s_{2}^{2}+35s_{2}^{3}\big{)}+s^{5}% \left(726s_{1}^{4}+2428s_{1}^{3}s_{2}-2469s_{1}^{2}s_{2}^{2}-35s_{2}^{4}\right% )+3s^{4}\big{(}242s_{1}^{5}-1850s_{1}^{4}s_{2}+925s_{1}^{3}s_{2}^{2}+581s_{1}^% {2}s_{2}^{3} \displaystyle-45s_{1}s_{2}^{4}+7s_{2}^{5}\big{)}-s^{3}(s_{1}-s_{2})\left(1052s% _{1}^{5}-1376s_{1}^{4}s_{2}-4151s_{1}^{3}s_{2}^{2}+705s_{1}^{2}s_{2}^{3}+101s_% {1}s_{2}^{4}-7s_{2}^{5}\right)+s^{2}(s_{1}-s_{2})^{3}\big{(}300s_{1}^{4}+1348s% _{1}^{3}s_{2} \displaystyle+675s_{1}^{2}s_{2}^{2}+24s_{1}s_{2}^{3}-s_{2}^{4}\big{)}+27ss_{1}% ^{2}(s_{1}-s_{2})^{5}(s_{1}+s_{2})-s_{1}^{2}(s_{1}-s_{2})^{7}\bigg{]}\bar{B}_{% 0}(s_{2})+\frac{16s^{2}s_{1}^{2}s_{2}^{2}}{\sigma^{6}}\bigg{[}3s^{6}-4s^{5}(s_% {1}+s_{2}) \displaystyle+s^{4}\left(-11s_{1}^{2}+40s_{1}s_{2}-11s_{2}^{2}\right)+12s^{3}(% 2s_{1}-s_{2})(s_{1}+s_{2})(s_{1}-2s_{2})-s^{2}\left(11s_{1}^{4}+36s_{1}^{3}s_{% 2}-108s_{1}^{2}s_{2}^{2}+36s_{1}s_{2}^{3}+11s_{2}^{4}\right) \displaystyle-4s(s_{1}-s_{2})^{2}(s_{1}+s_{2})\left(s_{1}^{2}-9s_{1}s_{2}+s_{2% }^{2}\right)+(s_{1}-s_{2})^{4}\left(3s_{1}^{2}+8s_{1}s_{2}+3s_{2}^{2}\right)% \bigg{]}C_{0}(s,s_{1},s_{2}) (G.1)

where \bar{B}_{0}(s_{i}) is defined as the regular part of the two point scalar integral. Then the others form factors in d=4 are explicitly given as follows. The expression of A_{2} is given by

 \displaystyle A_{2}^{Ren} \displaystyle=\frac{\pi^{2}}{450\sigma^{4}}\bigg{\{}n_{F}\bigg{[}7s^{8}+4(21s_% {1}+41s_{2})s^{7}-4\left(161s_{1}^{2}+815s_{2}s_{1}+211s_{2}^{2}\right)s^{6}+4% \left(427s_{1}^{3}+2199s_{2}s_{1}^{2}+819s_{2}^{2}s_{1}+367s_{2}^{3}\right)s^{% 5}-2\big{(}1155s_{1}^{4} \displaystyle+2850s_{2}s_{1}^{3}+19914s_{2}^{2}s_{1}^{2}-4430s_{2}^{3}s_{1}+45% 5s_{2}^{4}\big{)}s^{4}+4\left(427s_{1}^{5}-1425s_{2}s_{1}^{4}+18698s_{2}^{2}s_% {1}^{3}-1982s_{2}^{3}s_{1}^{2}-3645s_{2}^{4}s_{1}-73s_{2}^{5}\right)s^{3}+\big% {(}-644s_{1}^{6} \displaystyle+8796s_{2}s_{1}^{5}-39828s_{2}^{2}s_{1}^{4}-7928s_{2}^{3}s_{1}^{3% }+33972s_{2}^{4}s_{1}^{2}+4956s_{2}^{5}s_{1}+676s_{2}^{6}\big{)}s^{2}+4(s_{1}-% s_{2})^{3}\left(21s_{1}^{4}-752s_{2}s_{1}^{3}-1500s_{2}^{2}s_{1}^{2}-8s_{2}^{3% }s_{1}+79s_{2}^{4}\right)s \displaystyle+(s_{1}-s_{2})^{5}\left(7s_{1}^{3}+199s_{2}s_{1}^{2}+81s_{2}^{2}s% _{1}-47s_{2}^{3}\right)\bigg{]}+2n_{S}\bigg{[}-4s^{8}+(52s_{1}+42s_{2})s^{7}-% \left(232s_{1}^{2}+430s_{2}s_{1}+207s_{2}^{2}\right)s^{6}+2\big{(}262s_{1}^{3}% +519s_{2}s_{1}^{2} \displaystyle+864s_{2}^{2}s_{1}+277s_{2}^{3}\big{)}s^{5}-\left(680s_{1}^{4}+65% 0s_{2}s_{1}^{3}-591s_{2}^{2}s_{1}^{2}+2470s_{2}^{3}s_{1}+855s_{2}^{4}\right)s^% {4}+\big{(}524s_{1}^{5}-650s_{2}s_{1}^{4}-4224s_{2}^{2}s_{1}^{3}+6116s_{2}^{3}% s_{1}^{2}+460s_{2}^{4}s_{1} \displaystyle+774s_{2}^{5}\big{)}s^{3}+\left(-232s_{1}^{6}+1038s_{2}s_{1}^{5}+% 591s_{2}^{2}s_{1}^{4}+6116s_{2}^{3}s_{1}^{3}-8634s_{2}^{4}s_{1}^{2}+1518s_{2}^% {5}s_{1}-397s_{2}^{6}\right)s^{2}+2(s_{1}-s_{2})^{3}\big{(}26s_{1}^{4}-137s_{2% }s_{1}^{3} \displaystyle+375s_{2}^{2}s_{1}^{2}+327s_{2}^{3}s_{1}-51s_{2}^{4}\big{)}s-(s_{% 1}-s_{2})^{5}\left(4s_{1}^{3}-22s_{2}s_{1}^{2}+57s_{2}^{2}s_{1}-9s_{2}^{3}% \right)\bigg{]}+2n_{G}\bigg{[}487s^{8}-4(1039s_{1}+1044s_{2})s^{7}+2\big{(}759% 8s_{1}^{2} \displaystyle+12320s_{2}s_{1}+7623s_{2}^{2}\big{)}s^{6}-4\left(7793s_{1}^{3}+1% 2216s_{2}s_{1}^{2}+11871s_{2}^{2}s_{1}+7778s_{2}^{3}\right)s^{5}+\big{(}39290s% _{1}^{4}+28400s_{2}s_{1}^{3}+53202s_{2}^{2}s_{1}^{2}+24760s_{2}^{3}s_{1} \displaystyle+38940s_{2}^{4}\big{)}s^{4}-4\left(7793s_{1}^{5}-7100s_{2}s_{1}^{% 4}+10482s_{2}^{2}s_{1}^{3}+5312s_{2}^{3}s_{1}^{2}-7655s_{2}^{4}s_{1}+7668s_{2}% ^{5}\right)s^{3}+2\big{(}7598s_{1}^{6}-24432s_{2}s_{1}^{5}+26601s_{2}^{2}s_{1}% ^{4} \displaystyle-10624s_{2}^{3}s_{1}^{3}+17376s_{2}^{4}s_{1}^{2}-23952s_{2}^{5}s_% {1}+7433s_{2}^{6}\big{)}s^{2}-4(s_{1}-s_{2})^{3}\left(1039s_{1}^{4}-3043s_{2}s% _{1}^{3}-375s_{2}^{2}s_{1}^{2}+2853s_{2}^{3}s_{1}-1014s_{2}^{4}\right)s \displaystyle+(s_{1}-s_{2})^{5}\left(487s_{1}^{3}-1741s_{2}s_{1}^{2}+1671s_{2}% ^{2}s_{1}-477s_{2}^{3}\right)\bigg{]}\bigg{\}}-\frac{\pi^{2}s^{2}}{45\sigma^{5% }}\bigg{\{}n_{F}\bigg{[}7s^{8}-70(s_{1}+s_{2})s^{7}+\left(294s_{1}^{2}+490s_{2% }s_{1}+270s_{2}^{2}\right)s^{6}-2\big{(}343s_{1}^{3} \displaystyle+463s_{2}s_{1}^{2}+295s_{2}^{2}s_{1}+271s_{2}^{3}\big{)}s^{5}+10% \big{(}98s_{1}^{4}-27s_{2}s_{1}^{3}+774s_{2}^{2}s_{1}^{2}-99s_{2}^{3}s_{1}+62s% _{2}^{4}\big{)}s^{4}-2\big{(}441s_{1}^{5}-1495s_{2}s_{1}^{4}+9090s_{2}^{2}s_{1% }^{3}+258s_{2}^{3}s_{1}^{2} \displaystyle-1255s_{2}^{4}s_{1}+201s_{2}^{5}\big{)}s^{3}+10\big{(}49s_{1}^{6}% -397s_{2}s_{1}^{5}+995s_{2}^{2}s_{1}^{4}+2090s_{2}^{3}s_{1}^{3}-1633s_{2}^{4}s% _{1}^{2}-157s_{2}^{5}s_{1}+13s_{2}^{6}\big{)}s^{2}+2\big{(}-77s_{1}^{7}+1115s_% {2}s_{1}^{6}+2505s_{2}^{2}s_{1}^{5} \displaystyle-15015s_{2}^{3}s_{1}^{4}+8385s_{2}^{4}s_{1}^{3}+3057s_{2}^{5}s_{1% }^{2}+35s_{2}^{6}s_{1}-5s_{2}^{7}\big{)}s+3(s_{1}-s_{2})^{3}\big{(}7s_{1}^{5}-% 137s_{2}s_{1}^{4}-1832s_{2}^{2}s_{1}^{3}-1352s_{2}^{3}s_{1}^{2}-47s_{2}^{4}s_{% 1}+s_{2}^{5}\big{)}\bigg{]}-2n_{S}\bigg{[}-s^{8} \displaystyle+10(s_{1}+s_{2})s^{7}-\big{(}42s_{1}^{2}+70s_{2}s_{1}+45s_{2}^{2}% \big{)}s^{6}+2\big{(}49s_{1}^{3}+79s_{2}s_{1}^{2}+100s_{2}^{2}s_{1}+58s_{2}^{3% }\big{)}s^{5}-5\big{(}28s_{1}^{4}+18s_{2}s_{1}^{3}-9s_{2}^{2}s_{1}^{2}+36s_{2}% ^{3}s_{1}+37s_{2}^{4}\big{)}s^{4} \displaystyle+2\big{(}63s_{1}^{5}-85s_{2}s_{1}^{4}-270s_{2}^{2}s_{1}^{3}+834s_% {2}^{3}s_{1}^{2}-115s_{2}^{4}s_{1}+93s_{2}^{5}\big{)}s^{3}-5\big{(}14s_{1}^{6}% -62s_{2}s_{1}^{5}-5s_{2}^{2}s_{1}^{4}-380s_{2}^{3}s_{1}^{3}+652s_{2}^{4}s_{1}^% {2}-122s_{2}^{5}s_{1}+23s_{2}^{6}\big{)}s^{2} \displaystyle+2\big{(}11s_{1}^{7}-95s_{2}s_{1}^{6}+330s_{2}^{2}s_{1}^{5}-1680s% _{2}^{3}s_{1}^{4}+1245s_{2}^{4}s_{1}^{3}+399s_{2}^{5}s_{1}^{2}-230s_{2}^{6}s_{% 1}+20s_{2}^{7}\big{)}s-3(s_{1}-s_{2})^{3}\big{(}s_{1}^{5}-11s_{2}s_{1}^{4}+79s% _{2}^{2}s_{1}^{3}+319s_{2}^{3}s_{1}^{2}+34s_{2}^{4}s_{1} \displaystyle-2s_{2}^{5}\big{)}\bigg{]}-2n_{G}\bigg{[}13s^{8}-130(s_{1}+s_{2})% s^{7}+\big{(}546s_{1}^{2}+910s_{2}s_{1}+540s_{2}^{2}\big{)}s^{6}-2\big{(}637s_% {1}^{3}+937s_{2}s_{1}^{2}+895s_{2}^{2}s_{1}+619s_{2}^{3}\big{)}s^{5}+10\big{(}% 182s_{1}^{4}+27s_{2}s_{1}^{3} \displaystyle+747s_{2}^{2}s_{1}^{2}+9s_{2}^{3}s_{1}+173s_{2}^{4}\big{)}s^{4}-2% \big{(}819s_{1}^{5}-2005s_{2}s_{1}^{4}+7470s_{2}^{2}s_{1}^{3}+5262s_{2}^{3}s_{% 1}^{2}-1945s_{2}^{4}s_{1}+759s_{2}^{5}\big{)}s^{3}+10\big{(}91s_{1}^{6}-583s_{% 2}s_{1}^{5}+980s_{2}^{2}s_{1}^{4} \displaystyle+950s_{2}^{3}s_{1}^{3}+323s_{2}^{4}s_{1}^{2}-523s_{2}^{5}s_{1}+82% s_{2}^{6}\big{)}s^{2}+2\big{(}-143s_{1}^{7}+1685s_{2}s_{1}^{6}+525s_{2}^{2}s_{% 1}^{5}-4935s_{2}^{3}s_{1}^{4}+915s_{2}^{4}s_{1}^{3}+663s_{2}^{5}s_{1}^{2}+1415% s_{2}^{6}s_{1}-125s_{2}^{7}\big{)}s \displaystyle+3(s_{1}-s_{2})^{3}\big{(}13s_{1}^{5}-203s_{2}s_{1}^{4}-1358s_{2}% ^{2}s_{1}^{3}+562s_{2}^{3}s_{1}^{2}+157s_{2}^{4}s_{1}-11s_{2}^{5}\big{)}\bigg{% ]}\bigg{\}}\bar{B}_{0}(s)-\frac{\pi^{2}s_{1}^{2}}{45\sigma^{5}}\bigg{\{}n_{F}% \bigg{[}21s^{8}-2(77s_{1}+237s_{2})s^{7}+10\big{(}49s_{1}^{2} \displaystyle+223s_{2}s_{1}-420s_{2}^{2}\big{)}s^{6}+\big{(}-882s_{1}^{3}-3970% s_{2}s_{1}^{2}+5010s_{2}^{2}s_{1}+11178s_{2}^{3}\big{)}s^{5}+10\big{(}98s_{1}^% {4}+299s_{2}s_{1}^{3}+995s_{2}^{2}s_{1}^{2}-3003s_{2}^{3}s_{1}-405s_{2}^{4}% \big{)}s^{4} \displaystyle-2\big{(}343s_{1}^{5}+135s_{2}s_{1}^{4}+9090s_{2}^{2}s_{1}^{3}-10% 450s_{2}^{3}s_{1}^{2}-8385s_{2}^{4}s_{1}+3123s_{2}^{5}\big{)}s^{3}+2(s_{1}-s_{% 2})^{2}\big{(}147s_{1}^{4}-169s_{2}s_{1}^{3}+3385s_{2}^{2}s_{1}^{2}+6681s_{2}^% {3}s_{1}+1812s_{2}^{4}\big{)}s^{2} \displaystyle-10(s_{1}-s_{2})^{4}\big{(}7s_{1}^{3}-21s_{2}s_{1}^{2}-67s_{2}^{2% }s_{1}-15s_{2}^{3}\big{)}s+(s_{1}-s_{2})^{6}\big{(}7s_{1}^{2}-28s_{2}s_{1}-3s_% {2}^{2}\big{)}\bigg{]}-2n_{S}\bigg{[}-3s^{8}+(22s_{1}+42s_{2})s^{7}-5\big{(}14% s_{1}^{2}+38s_{2}s_{1}+69s_{2}^{2}\big{)}s^{6} \displaystyle+2\big{(}63s_{1}^{3}+155s_{2}s_{1}^{2}+330s_{2}^{2}s_{1}-72s_{2}^% {3}\big{)}s^{5}-5\big{(}28s_{1}^{4}+34s_{2}s_{1}^{3}-5s_{2}^{2}s_{1}^{2}+672s_% {2}^{3}s_{1}-405s_{2}^{4}\big{)}s^{4}+2\big{(}49s_{1}^{5}-45s_{2}s_{1}^{4}-270% s_{2}^{2}s_{1}^{3}+950s_{2}^{3}s_{1}^{2} \displaystyle+1245s_{2}^{4}s_{1}-1161s_{2}^{5}\big{)}s^{3}-(s_{1}-s_{2})^{2}% \big{(}42s_{1}^{4}-74s_{2}s_{1}^{3}-235s_{2}^{2}s_{1}^{2}-2064s_{2}^{3}s_{1}-6% 33s_{2}^{4}\big{)}s^{2}+10(s_{1}-s_{2})^{4}\big{(}s_{1}^{3}-3s_{2}s_{1}^{2}+2s% _{2}^{2}s_{1}+12s_{2}^{3}\big{)}s \displaystyle-(s_{1}-s_{2})^{6}\big{(}s_{1}^{2}-4s_{2}s_{1}+6s_{2}^{2}\big{)}% \bigg{]}-2n_{G}\bigg{[}39s^{8}-22(13s_{1}+33s_{2})s^{7}+10\big{(}91s_{1}^{2}+3% 37s_{2}s_{1}-213s_{2}^{2}\big{)}s^{6}-2\big{(}819s_{1}^{3}+2915s_{2}s_{1}^{2}-% 525s_{2}^{2}s_{1} \displaystyle-6021s_{2}^{3}\big{)}s^{5}+10\big{(}182s_{1}^{4}+401s_{2}s_{1}^{3% }+980s_{2}^{2}s_{1}^{2}-987s_{2}^{3}s_{1}-1620s_{2}^{4}\big{)}s^{4}+2\big{(}-6% 37s_{1}^{5}+135s_{2}s_{1}^{4}-7470s_{2}^{2}s_{1}^{3}+4750s_{2}^{3}s_{1}^{2}+91% 5s_{2}^{4}s_{1}+3843s_{2}^{5}\big{)}s^{3} \displaystyle+2(s_{1}-s_{2})^{2}\big{(}273s_{1}^{4}-391s_{2}s_{1}^{3}+2680s_{2% }^{2}s_{1}^{2}+489s_{2}^{3}s_{1}-87s_{2}^{4}\big{)}s^{2}-10(s_{1}-s_{2})^{4}% \big{(}13s_{1}^{3}-39s_{2}s_{1}^{2}-55s_{2}^{2}s_{1}+57s_{2}^{3}\big{)}s+(s_{1% }-s_{2})^{6}\big{(}13s_{1}^{2} \displaystyle-52s_{2}s_{1}+33s_{2}^{2}\big{)}\bigg{]}\bigg{\}}\bar{B}_{0}(s_{1% })+\frac{\pi^{2}s_{2}^{2}}{45\sigma^{5}}\bigg{\{}n_{F}\bigg{[}-45s^{8}+s^{7}(8% 10s_{1}+298s_{2})+10s^{6}\big{(}144s_{1}^{2}-295s_{1}s_{2}-85s_{2}^{2}\big{)}+% 2s^{5}\big{(}-6165s_{1}^{3}+5751s_{1}^{2}s_{2} \displaystyle+1745s_{1}s_{2}^{2}+681s_{2}^{3}\big{)}+10s^{4}\big{(}2025s_{1}^{% 4}-885s_{1}^{3}s_{2}-2003s_{1}^{2}s_{2}^{2}-59s_{1}s_{2}^{3}-134s_{2}^{4}\big{% )}-2s^{3}\big{(}6165s_{1}^{5}+4425s_{1}^{4}s_{2}-16910s_{1}^{3}s_{2}^{2}-354s_% {1}^{2}s_{2}^{3}+945s_{1}s_{2}^{4} \displaystyle-415s_{2}^{5}\big{)}+2s^{2}(s_{1}-s_{2})^{2}\big{(}720s_{1}^{4}+7% 191s_{1}^{3}s_{2}+3647s_{1}^{2}s_{2}^{2}+457s_{1}s_{2}^{3}-159s_{2}^{4}\big{)}% +10s(s_{1}-s_{2})^{4}\big{(}81s_{1}^{3}+29s_{1}^{2}s_{2}-21s_{1}s_{2}^{2}+7s_{% 2}^{3}\big{)}-(s_{1}-s_{2})^{6}\big{(}45s_{1}^{2} \displaystyle-28s_{1}s_{2}+7s_{2}^{2}\big{)}\bigg{]}+2n_{G}\bigg{[}45s^{8}-2s^% {7}(405s_{1}+161s_{2})+s^{6}\big{(}-1440s_{1}^{2}+3550s_{1}s_{2}+1000s_{2}^{2}% \big{)}+2s^{5}\big{(}6165s_{1}^{3}-1539s_{1}^{2}s_{2}-2855s_{1}s_{2}^{2}-879s_% {2}^{3}\big{)}
 \displaystyle-10s^{4}\big{(}2025s_{1}^{4}+15s_{1}^{3}s_{2}-1232s_{1}^{2}s_{2}^% {2}-341s_{1}s_{2}^{3}-191s_{2}^{4}\big{)}+2s^{3}\big{(}6165s_{1}^{5}-75s_{1}^{% 4}s_{2}-2090s_{1}^{3}s_{2}^{2}-5286s_{1}^{2}s_{2}^{3}+405s_{1}s_{2}^{4}-655s_{% 2}^{5}\big{)} \displaystyle-2s^{2}(s_{1}-s_{2})^{2}\big{(}720s_{1}^{4}+2979s_{1}^{3}s_{2}-92% 2s_{1}^{2}s_{2}^{2}+463s_{1}s_{2}^{3}-276s_{2}^{4}\big{)}-10s(s_{1}-s_{2})^{4}% \big{(}81s_{1}^{3}-31s_{1}^{2}s_{2}-39s_{1}s_{2}^{2}+13s_{2}^{3}\big{)}+(s_{1}% -s_{2})^{6}\big{(}45s_{1}^{2}-52s_{1}s_{2} \displaystyle+13s_{2}^{2}\big{)}\bigg{]}+2n_{S}s_{2}\bigg{[}4s^{7}-25s^{6}(4s_% {1}+s_{2})+s^{5}\big{(}-1404s_{1}^{2}+370s_{1}s_{2}+66s_{2}^{2}\big{)}+5s^{4}% \big{(}300s_{1}^{3}+257s_{1}^{2}s_{2}-94s_{1}s_{2}^{2}-19s_{2}^{3}\big{)}+4s^{% 3}\big{(}375s_{1}^{4}-1235s_{1}^{3}s_{2} \displaystyle+411s_{1}^{2}s_{2}^{2}+45s_{1}s_{2}^{3}+20s_{2}^{4}\big{)}-s^{2}(% s_{1}-s_{2})^{2}\big{(}1404s_{1}^{3}+1523s_{1}^{2}s_{2}-2s_{1}s_{2}^{2}+39s_{2% }^{3}\big{)}-10s(s_{1}-s_{2})^{4}\big{(}10s_{1}^{2}+3s_{1}s_{2}-s_{2}^{2}\big{% )}+(s_{1}-s_{2})^{6}(4s_{1}-s_{2})\bigg{]}\bigg{\}}\bar{B}_{0}(s_{2}) \displaystyle-\frac{4\pi^{2}s^{2}s_{1}^{2}s_{2}^{2}}{3\sigma^{5}}\bigg{\{}n_{F% }\bigg{[}45s^{5}-3s^{4}(45s_{1}+13s_{2})+2s^{3}\big{(}45s_{1}^{2}+152s_{1}s_{2% }-63s_{2}^{2}\big{)}+2s^{2}\big{(}45s_{1}^{3}-265s_{1}^{2}s_{2}+59s_{1}s_{2}^{% 2}+81s_{2}^{3}\big{)}+s\big{(}-135s_{1}^{4}+304s_{1}^{3}s_{2} \displaystyle+118s_{1}^{2}s_{2}^{2}-272s_{1}s_{2}^{3}-15s_{2}^{4}\big{)}+3(s_{% 1}-s_{2})^{3}\big{(}15s_{1}^{2}+32s_{1}s_{2}+9s_{2}^{2}\big{)}\bigg{]}-2n_{G}% \bigg{[}45s^{5}-3s^{4}(45s_{1}+37s_{2})+s^{3}\big{(}90s_{1}^{2}+256s_{1}s_{2}+% 36s_{2}^{2}\big{)}+2s^{2}\big{(}45s_{1}^{3} \displaystyle-145s_{1}^{2}s_{2}-64s_{1}s_{2}^{2}+54s_{2}^{3}\big{)}+s\big{(}-1% 35s_{1}^{4}+256s_{1}^{3}s_{2}-128s_{1}^{2}s_{2}^{2}+112s_{1}s_{2}^{3}-105s_{2}% ^{4}\big{)}+3(s_{1}-s_{2})^{3}\big{(}15s_{1}^{2}+8s_{1}s_{2}-9s_{2}^{2}\big{)}% \bigg{]}-2n_{S}s_{2}\bigg{[}12s^{4}+s^{3}(8s_{1} \displaystyle-27s_{2})+s^{2}\big{(}-40s_{1}^{2}+41s_{1}s_{2}+9s_{2}^{2}\big{)}% +s\big{(}8s_{1}^{3}+41s_{1}^{2}s_{2}-64s_{1}s_{2}^{2}+15s_{2}^{3}\big{)}+3(s_{% 1}-s_{2})^{3}(4s_{1}+3s_{2})\bigg{]}\bigg{\}}C_{0}(s,s_{1},s_{2}). (G.2)

The form factor A_{3} is given by

 \displaystyle A_{3}^{Ren} \displaystyle=\frac{\pi^{2}}{900\sigma^{3}}\bigg{\{}s_{2}^{5}\big{[}s^{2}(763n% _{F}+12166n_{G}-22n_{S})-6ss_{1}(17n_{F}-2956n_{G}+77n_{S})+s_{1}^{2}(763n_{F}% +12166n_{G}-22n_{S})\big{]}-2s_{2}^{4}(s+s_{1})\big{[}5s^{2}(24n_{F} \displaystyle+1988n_{G}-31n_{S})+ss_{1}(-2414n_{F}+2202n_{G}-659n_{S})+5s_{1}^% {2}(24n_{F}+1988n_{G}-31n_{S})\big{]}+s_{2}^{2}(s-s_{1})^{2}(s+s_{1})\big{[}s^% {2}(896n_{F}-10678n_{G} \displaystyle+451n_{S})+4ss_{1}(941n_{F}+1112n_{G}+196n_{S})+s_{1}^{2}(896n_{F% }-10678n_{G}+451n_{S})\big{]}-s_{2}(s-s_{1})^{4}\big{[}s^{2}(423n_{F}-3214n_{G% }+188n_{S})+2ss_{1}(551n_{F} \displaystyle-918n_{G}+206n_{S})+s_{1}^{2}(423n_{F}-3214n_{G}+188n_{S})\big{]}% -s_{2}^{3}\big{[}5s^{4}(139n_{F}-3812n_{G}+109n_{S})+2s^{3}s_{1}(3556n_{F}-225% 8n_{G}+811n_{S})+2s^{2}s_{1}^{2}(641n_{F} \displaystyle-1888n_{G}+1021n_{S})+2ss_{1}^{3}(3556n_{F}-2258n_{G}+811n_{S})+5% s_{1}^{4}(139n_{F}-3812n_{G}+109n_{S})\big{]}-s_{2}^{6}(s+s_{1})(472n_{F}+4054% n_{G}+57n_{S}) \displaystyle+4(s-s_{1})^{6}(s+s_{1})(18n_{F}-99n_{G}+8n_{S})+s_{2}^{7}(99n_{F% }+568n_{G}+19n_{S})\bigg{\}}+\frac{\pi^{2}s^{2}}{90\sigma^{4}}\bigg{\{}-s_{2}^% {5}\big{[}s^{2}(443n_{F}+386n_{G}+103n_{S})+40ss_{1}(26n_{F} \displaystyle+7(2n_{G}+n_{S}))+3s_{1}^{2}(259n_{F}+358n_{G}-11n_{S})\big{]}+2s% _{2}^{2}(s-s_{1})^{3}\big{[}s^{2}(208n_{F}+301n_{G}+38n_{S})+4ss_{1}(112n_{F}+% 214n_{G}+17n_{S})+3s_{1}^{2}(154n_{F} \displaystyle-17n_{G}+19n_{S})\big{]}+s_{2}^{4}\big{[}5s^{3}(146n_{F}+152n_{G}% +31n_{S})+s^{2}s_{1}(1642n_{F}+1384n_{G}+347n_{S})+ss_{1}^{2}(-14n_{F}+1852n_{% G}+191n_{S})+15s_{1}^{3}(22n_{F} \displaystyle+196n_{G}-23n_{S})\big{]}-s_{2}^{3}(s-s_{1})\big{[}5s^{3}(143n_{F% }+176n_{G}+28n_{S})+5s^{2}s_{1}(311n_{F}+52(8n_{G}+n_{S}))+ss_{1}^{2}(181n_{F}% +292n_{G}+386n_{S})+15s_{1}^{3}(67n_{F} \displaystyle-164n_{G}+22n_{S})\big{]}+2s_{2}^{6}(74n_{F}s+114n_{F}s_{1}+53n_{% G}s+33n_{G}s_{1}+19n_{S}s+39n_{S}s_{1})-s_{2}(s-s_{1})^{5}(s(133n_{F}+226n_{G}% +23n_{S})+3s_{1}(59n_{F} \displaystyle+158n_{G}+9n_{S}))+3(s-s_{1})^{7}(6n_{F}+12n_{G}+n_{S})-3s_{2}^{7% }(7n_{F}+4n_{G}+2n_{S})\bigg{\}}\bar{B}_{0}(s)+\frac{\pi^{2}s_{1}^{2}}{90% \sigma^{4}}\bigg{\{}-s_{2}^{5}\big{[}3s^{2}(259n_{F}+358n_{G}-11n_{S})+40ss_{1} \displaystyle\times(26n_{F}+7(2n_{G}+n_{S}))+s_{1}^{2}(443n_{F}+386n_{G}+103n_% {S})\big{]}-2s_{2}^{2}(s-s_{1})^{3}\big{[}3s^{2}(154n_{F}-17n_{G}+19n_{S})+4ss% _{1}(112n_{F}+214n_{G}+17n_{S}) \displaystyle+s_{1}^{2}(208n_{F}+301n_{G}+38n_{S})\big{]}+s_{2}^{4}\big{[}15s^% {3}(22n_{F}+196n_{G}-23n_{S})+s^{2}s_{1}(-14n_{F}+1852n_{G}+191n_{S})+ss_{1}^{% 2}(1642n_{F}+1384n_{G}+347n_{S}) \displaystyle+5s_{1}^{3}(146n_{F}+152n_{G}+31n_{S})\big{]}+s_{2}^{3}(s-s_{1})% \big{[}15s^{3}(67n_{F}-164n_{G}+22n_{S})+s^{2}s_{1}(181n_{F}+292n_{G}+386n_{S}% )+5ss_{1}^{2}(311n_{F}+52(8n_{G}+n_{S})) \displaystyle+5s_{1}^{3}(143n_{F}+176n_{G}+28n_{S})\big{]}+2s_{2}^{6}(114n_{F}% s+74n_{F}s_{1}+33n_{G}s+53n_{G}s_{1}+39n_{S}s+19n_{S}s_{1})+s_{2}(s-s_{1})^{5}% (3s(59n_{F}+158n_{G}+9n_{S}) \displaystyle+s_{1}(133n_{F}+226n_{G}+23n_{S}))-3(s-s_{1})^{7}(6n_{F}+12n_{G}+% n_{S})-3s_{2}^{7}(7n_{F}+4n_{G}+2n_{S})\bigg{\}}\bar{B}_{0}(s_{1})+\frac{\pi^{% 2}s_{2}^{2}}{90\sigma^{4}}\bigg{\{}s_{2}^{5}\big{[}5s^{2}(37n_{F}+292n_{G}+2n_% {S}) \displaystyle+8ss_{1}(19n_{F}+238n_{G}-n_{S})+5s_{1}^{2}(37n_{F}+292n_{G}+2n_{% S})\big{]}-2s_{2}^{4}(s+s_{1})\big{[}5s^{2}(26n_{F}+257n_{G}+n_{S})-4ss_{1}(58% n_{F}+61n_{G}+8n_{S})+5s_{1}^{2}(26n_{F} \displaystyle+257n_{G}+n_{S})\big{]}+s_{2}^{3}\big{[}5s^{4}(41n_{F}+542n_{G}+n% _{S})-8s^{3}s_{1}(103n_{F}+196n_{G}+8n_{S})-6s^{2}s_{1}^{2}(107n_{F}+134n_{G}+% 77n_{S})-8ss_{1}^{3}(103n_{F}+196n_{G} \displaystyle+8n_{S})+5s_{1}^{4}(41n_{F}+542n_{G}+n_{S})\big{]}-s_{2}^{2}(s+s_% {1})\big{[}s^{4}(86n_{F}+1712n_{G}+n_{S})-24s^{3}s_{1}(36n_{F}+282n_{G}+n_{S})% -2s^{2}s_{1}^{2}(182n_{F}-3916n_{G} \displaystyle+187n_{S})-24ss_{1}^{3}(36n_{F}+282n_{G}+n_{S})+s_{1}^{4}(86n_{F}% +1712n_{G}+n_{S})\big{]}-5s_{2}^{6}(s+s_{1})(14n_{F}+92n_{G}+n_{S})+s_{2}^{7}(% 11n_{F}+62n_{G}+n_{S}) \displaystyle+15s_{2}(s-s_{1})^{2}\big{[}s^{4}(n_{F}+40n_{G})-2s^{3}s_{1}(7n_{% F}+88n_{G})-2s^{2}s_{1}^{2}(47n_{F}+32n_{G})-2ss_{1}^{3}(7n_{F}+88n_{G})+s_{1}% ^{4}(n_{F}+40n_{G})\big{]}-90n_{G}(s-s_{1})^{4}(s+s_{1}) \displaystyle\times\big{[}s^{2}-8ss_{1}+s_{1}^{2}\big{]}\bigg{\}}\bar{B}_{0}(s% _{2})+\frac{2\pi^{2}s^{2}s_{1}^{2}s_{2}^{2}}{3\sigma^{4}}\bigg{\{}-s_{2}^{2}% \big{[}3s^{2}(11n_{F}+8n_{G}+n_{S})-2ss_{1}(n_{F}-14n_{G}-4n_{S})+3s_{1}^{2}(1% 1n_{F}+8n_{G}+n_{S})\big{]} \displaystyle+3s_{2}^{3}(s+s_{1})(7n_{F}-2n_{G}+2n_{S})-3s_{2}^{4}(n_{F}-2n_{G% }+n_{S})+3s_{2}(5n_{F}+14n_{G})(s+s_{1})(s-s_{1})^{2}-18n_{G}(s-s_{1})^{4}% \bigg{\}}C_{0}(s,s_{1},s_{2}) \displaystyle-\frac{8\pi^{2}}{720}\,(s+s_{1}+s_{2})(n_{S}+11n_{F}+62n_{G}). (G.3)

Then A_{4} is expressed as

 \displaystyle A_{4}^{Ren} \displaystyle=\frac{\pi^{2}}{900\sigma^{3}}\bigg{\{}n_{F}\bigg{[}151s^{7}-s^{6% }(675s_{1}+629s_{2})+s^{5}\big{(}1119s_{1}^{2}-1350s_{1}s_{2}+743s_{2}^{2}\big% {)}+s^{4}\big{(}-595s_{1}^{3}+7221s_{1}^{2}s_{2}+8339s_{1}s_{2}^{2}+435s_{2}^{% 3}\big{)}-s^{3}\big{(}595s_{1}^{4} \displaystyle+10484s_{1}^{3}s_{2}+13882s_{1}^{2}s_{2}^{2}+9356s_{1}s_{2}^{3}+1% 875s_{2}^{4}\big{)}+s^{2}\big{(}1119s_{1}^{5}+7221s_{1}^{4}s_{2}-13882s_{1}^{3% }s_{2}^{2}+2034s_{1}^{2}s_{2}^{3}+1659s_{1}s_{2}^{4}+1849s_{2}^{5}\big{)}-s(s_% {1}-s_{2})^{3}\big{(}675s_{1}^{3} \displaystyle+3375s_{1}^{2}s_{2}-239s_{1}s_{2}^{2}-811s_{2}^{3}\big{)}+(s_{1}-% s_{2})^{5}\big{(}151s_{1}^{2}+126s_{1}s_{2}-137s_{2}^{2}\big{)}\bigg{]}+2n_{G}% \bigg{[}91s^{7}-s^{6}(475s_{1}+1689s_{2})+s^{5}\big{(}879s_{1}^{2}+7650s_{1}s_% {2}+7963s_{2}^{2}\big{)} \displaystyle-s^{4}\big{(}495s_{1}^{3}+14439s_{1}^{2}s_{2}+15401s_{1}s_{2}^{2}% +17665s_{2}^{3}\big{)}+s^{3}\big{(}-495s_{1}^{4}+16956s_{1}^{3}s_{2}+8638s_{1}% ^{2}s_{2}^{2}-596s_{1}s_{2}^{3}+21625s_{2}^{4}\big{)}+s^{2}\big{(}879s_{1}^{5}% -14439s_{1}^{4}s_{2} \displaystyle+8638s_{1}^{3}s_{2}^{2}-4806s_{1}^{2}s_{2}^{3}+24819s_{1}s_{2}^{4% }-15091s_{2}^{5}\big{)}-s(s_{1}-s_{2})^{3}\big{(}475s_{1}^{3}-6225s_{1}^{2}s_{% 2}-4699s_{1}s_{2}^{2}+5649s_{2}^{3}\big{)}+(s_{1}-s_{2})^{5}\big{(}91s_{1}^{2}% -1234s_{1}s_{2}+883s_{2}^{2}\big{)}\bigg{]} \displaystyle+4n_{S}\bigg{[}14s^{7}-s^{6}(75s_{1}+56s_{2})+s^{5}\big{(}141s_{1% }^{2}+52s_{2}^{2}\big{)}+s^{4}\big{(}-80s_{1}^{3}+744s_{1}^{2}s_{2}+496s_{1}s_% {2}^{2}+90s_{2}^{3}\big{)}-2s^{3}\big{(}40s_{1}^{4}+688s_{1}^{3}s_{2}+124s_{1}% ^{2}s_{2}^{2}+317s_{1}s_{2}^{3} \displaystyle+125s_{2}^{4}\big{)}+s^{2}\big{(}141s_{1}^{5}+744s_{1}^{4}s_{2}-2% 48s_{1}^{3}s_{2}^{2}-924s_{1}^{2}s_{2}^{3}+51s_{1}s_{2}^{4}+236s_{2}^{5}\big{)% }-s(s_{1}-s_{2})^{3}\big{(}75s_{1}^{3}+225s_{1}^{2}s_{2}-46s_{1}s_{2}^{2}-104s% _{2}^{3}\big{)}+2(s_{1}-s_{2})^{5}\big{(}7s_{1}^{2} \displaystyle+7s_{1}s_{2}-9s_{2}^{2}\big{)}\bigg{]}\bigg{\}}+\frac{\pi^{2}s^{2% }}{90\sigma^{4}}\bigg{\{}n_{F}\bigg{[}29s^{7}-7s^{6}(29s_{1}+27s_{2})+s^{5}% \big{(}625s_{1}^{2}+584s_{1}s_{2}+525s_{2}^{2}\big{)}-s^{4}\big{(}1095s_{1}^{3% }+309s_{1}^{2}s_{2}+231s_{1}s_{2}^{2}+805s_{2}^{3}\big{)} \displaystyle+s^{3}\big{(}1175s_{1}^{4}-1272s_{1}^{3}s_{2}-3822s_{1}^{2}s_{2}^% {2}-856s_{1}s_{2}^{3}+735s_{2}^{4}\big{)}+s^{2}\big{(}-769s_{1}^{5}+2665s_{1}^% {4}s_{2}+978s_{1}^{3}s_{2}^{2}+4526s_{1}^{2}s_{2}^{3}+1159s_{1}s_{2}^{4}-399s_% {2}^{5}\big{)}+s\big{(}283s_{1}^{6} \displaystyle-2064s_{1}^{5}s_{2}+2625s_{1}^{4}s_{2}^{2}-960s_{1}^{3}s_{2}^{3}+% 525s_{1}^{2}s_{2}^{4}-528s_{1}s_{2}^{5}+119s_{2}^{6}\big{)}-15(s_{1}-s_{2})^{3% }\big{(}3s_{1}^{4}-30s_{1}^{3}s_{2}-94s_{1}^{2}s_{2}^{2}+2s_{1}s_{2}^{3}-s_{2}% ^{4}\big{)}\bigg{]}+2n_{G}\bigg{[}49s^{7} \displaystyle-s^{6}(343s_{1}+369s_{2})+s^{5}\big{(}1025s_{1}^{2}+1384s_{1}s_{2% }+1185s_{2}^{2}\big{)}-s^{4}\big{(}1695s_{1}^{3}+1329s_{1}^{2}s_{2}+1191s_{1}s% _{2}^{2}+2105s_{2}^{3}\big{)}+s^{3}\big{(}1675s_{1}^{4}-1032s_{1}^{3}s_{2}-822% s_{1}^{2}s_{2}^{2} \displaystyle-2216s_{1}s_{2}^{3}+2235s_{2}^{4}\big{)}+s^{2}\big{(}-989s_{1}^{5% }+2765s_{1}^{4}s_{2}-582s_{1}^{3}s_{2}^{2}+646s_{1}^{2}s_{2}^{3}+5099s_{1}s_{2% }^{4}-1419s_{2}^{5}\big{)}+s\big{(}323s_{1}^{6}-1824s_{1}^{5}s_{2}+3765s_{1}^{% 4}s_{2}^{2}-480s_{1}^{3}s_{2}^{3} \displaystyle+1365s_{1}^{2}s_{2}^{4}-3648s_{1}s_{2}^{5}+499s_{2}^{6}\big{)}-15% (s_{1}-s_{2})^{3}\big{(}3s_{1}^{4}-18s_{1}^{3}s_{2}+94s_{1}^{2}s_{2}^{2}+46s_{% 1}s_{2}^{3}-5s_{2}^{4}\big{)}\bigg{]}+4n_{S}s\bigg{[}s^{6}-s^{5}(7s_{1}+6s_{2}% )+s^{4}\big{(}20s_{1}^{2}+16s_{1}s_{2}+15s_{2}^{2}\big{)} \displaystyle+s^{3}\big{(}-30s_{1}^{3}+24s_{1}^{2}s_{2}+6s_{1}s_{2}^{2}-20s_{2% }^{3}\big{)}+s^{2}\big{(}25s_{1}^{4}-108s_{1}^{3}s_{2}-18s_{1}^{2}s_{2}^{2}-44% s_{1}s_{2}^{3}+15s_{2}^{4}\big{)}+s\big{(}-11s_{1}^{5}+110s_{1}^{4}s_{2}+282s_% {1}^{3}s_{2}^{2}-116s_{1}^{2}s_{2}^{3}+41s_{1}s_{2}^{4} \displaystyle-6s_{2}^{5}\big{)}+2s_{1}^{6}-36s_{1}^{5}s_{2}-285s_{1}^{4}s_{2}^% {2}+240s_{1}^{3}s_{2}^{3}+90s_{1}^{2}s_{2}^{4}-12s_{1}s_{2}^{5}+s_{2}^{6}\bigg% {]}\bigg{\}}\bar{B}_{0}(s)+\frac{\pi^{2}s_{1}^{2}}{90\sigma^{4}}\bigg{\{}-n_{F% }\bigg{[}45s^{7}-s^{6}(283s_{1}+585s_{2})+s^{5}\big{(}769s_{1}^{2}+2064s_{1}s_% {2} \displaystyle+75s_{2}^{2}\big{)}-5s^{4}\big{(}235s_{1}^{3}+533s_{1}^{2}s_{2}+5% 25s_{1}s_{2}^{2}-573s_{2}^{3}\big{)}+3s^{3}\big{(}365s_{1}^{4}+424s_{1}^{3}s_{% 2}-326s_{1}^{2}s_{2}^{2}+320s_{1}s_{2}^{3}-1295s_{2}^{4}\big{)}-s^{2}(s_{1}-s_% {2})^{2}\big{(}625s_{1}^{3}+941s_{1}^{2}s_{2} \displaystyle-2565s_{1}s_{2}^{2}-1545s_{2}^{3}\big{)}+s(s_{1}-s_{2})^{4}\big{(% }203s_{1}^{2}+228s_{1}s_{2}-75s_{2}^{2}\big{)}-(29s_{1}-15s_{2})(s_{1}-s_{2})^% {6}\bigg{]}-2n_{G}\bigg{[}45s^{7}-s^{6}(323s_{1}+405s_{2})+s^{5}\big{(}989s_{1% }^{2} \displaystyle+1824s_{1}s_{2}+2355s_{2}^{2}\big{)}-5s^{4}\big{(}335s_{1}^{3}+55% 3s_{1}^{2}s_{2}+753s_{1}s_{2}^{2}+879s_{2}^{3}\big{)}+3s^{3}\big{(}565s_{1}^{4% }+344s_{1}^{3}s_{2}+194s_{1}^{2}s_{2}^{2}+160s_{1}s_{2}^{3}+785s_{2}^{4}\big{)% }-s^{2}(s_{1}-s_{2})^{2} \displaystyle\times\big{(}1025s_{1}^{3}+721s_{1}^{2}s_{2}-405s_{1}s_{2}^{2}-88% 5s_{2}^{3}\big{)}+s(s_{1}-s_{2})^{4}\big{(}343s_{1}^{2}-12s_{1}s_{2}-915s_{2}^% {2}\big{)}-(49s_{1}-75s_{2})(s_{1}-s_{2})^{6}\bigg{]}+4n_{S}s_{1}\bigg{[}2s^{6% }-s^{5}(11s_{1}+36s_{2}) \displaystyle+5s^{4}\big{(}5s_{1}^{2}+22s_{1}s_{2}-57s_{2}^{2}\big{)}-6s^{3}% \big{(}5s_{1}^{3}+18s_{1}^{2}s_{2}-47s_{1}s_{2}^{2}-40s_{2}^{3}\big{)}+2s^{2}(% s_{1}-s_{2})^{2}\big{(}10s_{1}^{2}+32s_{1}s_{2}+45s_{2}^{2}\big{)}-s(s_{1}-s_{% 2})^{4}(7s_{1}+12s_{2}) \displaystyle+(s_{1}-s_{2})^{6}\bigg{]}\bigg{\}}\bar{B}_{0}(s_{1})+\frac{\pi^{% 2}s_{2}^{2}}{90\sigma^{4}}\bigg{\{}2n_{G}\bigg{[}3s^{7}+5s^{6}(s_{2}-9s_{1})-s% ^{5}\big{(}243s_{1}^{2}+224s_{1}s_{2}+93s_{2}^{2}\big{)}+s^{4}\big{(}285s_{1}^% {3}-1469s_{1}^{2}s_{2}+1211s_{1}s_{2}^{2}+285s_{2}^{3}\big{)} \displaystyle+s^{3}\big{(}285s_{1}^{4}+2896s_{1}^{3}s_{2}+5082s_{1}^{2}s_{2}^{% 2}-1704s_{1}s_{2}^{3}-415s_{2}^{4}\big{)}-s^{2}(s_{1}-s_{2})^{2}\big{(}243s_{1% }^{3}+1955s_{1}^{2}s_{2}-1415s_{1}s_{2}^{2}-327s_{2}^{3}\big{)}-s(s_{1}-s_{2})% ^{4}\big{(}45s_{1}^{2} \displaystyle+404s_{1}s_{2}+135s_{2}^{2}\big{)}+(s_{1}-s_{2})^{6}(3s_{1}+23s_{% 2})\bigg{]}-4n_{S}\bigg{[}3s^{7}-10s^{6}(3s_{1}+2s_{2})+s^{5}\big{(}162s_{1}^{% 2}+116s_{1}s_{2}+57s_{2}^{2}\big{)}-s^{4}\big{(}135s_{1}^{3}+244s_{1}^{2}s_{2}% +149s_{1}s_{2}^{2} \displaystyle+90s_{2}^{3}\big{)}+s^{3}\big{(}-135s_{1}^{4}+416s_{1}^{3}s_{2}-1% 8s_{1}^{2}s_{2}^{2}+36s_{1}s_{2}^{3}+85s_{2}^{4}\big{)}+2s^{2}(s_{1}-s_{2})^{2% }\big{(}81s_{1}^{3}+40s_{1}^{2}s_{2}-10s_{1}s_{2}^{2}-24s_{2}^{3}\big{)}-s(s_{% 1}-s_{2})^{4}\big{(}30s_{1}^{2}+4s_{1}s_{2} \displaystyle-15s_{2}^{2}\big{)}+(s_{1}-s_{2})^{6}(3s_{1}-2s_{2})\bigg{]}-n_{F% }\big{(}57s^{7}-5s^{6}(135s_{1}+77s_{2})+s^{5}\big{(}243s_{1}^{2}+2704s_{1}s_{% 2}+1113s_{2}^{2}\big{)}+s^{4}\big{(}375s_{1}^{3}-1511s_{1}^{2}s_{2}-3751s_{1}s% _{2}^{2} \displaystyle-1785s_{2}^{3}\big{)}+s^{3}\big{(}375s_{1}^{4}-3536s_{1}^{3}s_{2}% +1518s_{1}^{2}s_{2}^{2}+1464s_{1}s_{2}^{3}+1715s_{2}^{4}\big{)}+s^{2}(s_{1}-s_% {2})^{2}\big{(}243s_{1}^{3}-1025s_{1}^{2}s_{2}-775s_{1}s_{2}^{2}-987s_{2}^{3}% \big{)}-s(s_{1}-s_{2})^{4} \displaystyle\times\big{(}675s_{1}^{2}-4s_{1}s_{2}-315s_{2}^{2}\big{)}+(s_{1}-% s_{2})^{6}(57s_{1}-43s_{2})\big{)}\bigg{\}}\bar{B}_{0}(s_{2})-\frac{2\pi^{2}s^% {2}s_{1}^{2}s_{2}^{2}}{3\sigma^{4}}\bigg{\{}n_{F}\bigg{[}21s^{4}-18s^{3}(2s_{1% }+3s_{2})+s^{2}\big{(}30s_{1}^{2}+70s_{1}s_{2}+36s_{2}^{2}\big{)} \displaystyle+s\big{(}-36s_{1}^{3}+70s_{1}^{2}s_{2}-40s_{1}s_{2}^{2}+6s_{2}^{3% }\big{)}+3(s_{1}-s_{2})^{3}(7s_{1}+3s_{2})\bigg{]}-2n_{G}\bigg{[}3s^{4}+18s^{3% }s_{2}-2s^{2}\big{(}3s_{1}^{2}+7s_{1}s_{2}+36s_{2}^{2}\big{)}+2ss_{2}\big{(}-7% s_{1}^{2}-32s_{1}s_{2}+39s_{2}^{2}\big{)} \displaystyle+3(s_{1}-s_{2})^{3}(s_{1}+9s_{2})\big{]}-4n_{S}ss_{1}\bigg{[}3s^{% 2}+s(s_{2}-6s_{1})+3s_{1}^{2}+s_{1}s_{2}-4s_{2}^{2}\bigg{]}\bigg{\}}C_{0}(s,s_% {1},s_{2})-\frac{16\pi^{2}}{720}\,(s+s_{1}+s_{2})\big{(}n_{S}+11n_{F}+62n_{G}% \big{)}. (G.4)

Finally the A_{5} form factor can be written as

 \displaystyle A_{5}^{Ren} \displaystyle=\frac{\pi^{2}}{1800\sigma^{2}}\bigg{\{}n_{F}\bigg{[}137s^{6}-674% s^{5}(s_{1}+s_{2})+11s^{4}\big{(}133s_{1}^{2}+150s_{1}s_{2}+133s_{2}^{2}\big{)% }-4s^{3}\big{(}463s_{1}^{3}+244s_{1}^{2}s_{2}+244s_{1}s_{2}^{2}+463s_{2}^{3}% \big{)}+s^{2}\big{(}1463s_{1}^{4} \displaystyle-976s_{1}^{3}s_{2}+1234s_{1}^{2}s_{2}^{2}-976s_{1}s_{2}^{3}+1463s% _{2}^{4}\big{)}-2s(s_{1}-s_{2})^{2}\big{(}337s_{1}^{3}-151s_{1}^{2}s_{2}-151s_% {1}s_{2}^{2}+337s_{2}^{3}\big{)}+(s_{1}-s_{2})^{4}\big{(}137s_{1}^{2}-126s_{1}% s_{2}+137s_{2}^{2}\big{)}\bigg{]} \displaystyle-2n_{G}\bigg{[}883s^{6}-4766s^{5}(s_{1}+s_{2})+s^{4}\big{(}11117s% _{1}^{2}+12450s_{1}s_{2}+11117s_{2}^{2}\big{)}-4s^{3}\big{(}3617s_{1}^{3}+1921% s_{1}^{2}s_{2}+1921s_{1}s_{2}^{2}+3617s_{2}^{3}\big{)}+s^{2}\big{(}11117s_{1}^% {4} \displaystyle-7684s_{1}^{3}s_{2}-194s_{1}^{2}s_{2}^{2}-7684s_{1}s_{2}^{3}+1111% 7s_{2}^{4}\big{)}-2s(s_{1}-s_{2})^{2}\big{(}2383s_{1}^{3}-1459s_{1}^{2}s_{2}-1% 459s_{1}s_{2}^{2}+2383s_{2}^{3}\big{)}+(s_{1}-s_{2})^{4}\big{(}883s_{1}^{2}-12% 34s_{1}s_{2} \displaystyle+883s_{2}^{2}\big{)}\bigg{]}+4n_{S}\bigg{[}18s^{6}-86s^{5}(s_{1}+% s_{2})+7s^{4}\big{(}26s_{1}^{2}+25s_{1}s_{2}+26s_{2}^{2}\big{)}-s^{3}\big{(}22% 8s_{1}^{3}+89s_{1}^{2}s_{2}+89s_{1}s_{2}^{2}+228s_{2}^{3}\big{)}+s^{2}\big{(}1% 82s_{1}^{4} \displaystyle-89s_{1}^{3}s_{2}+26s_{1}^{2}s_{2}^{2}-89s_{1}s_{2}^{3}+182s_{2}^% {4}\big{)}-s(s_{1}-s_{2})^{2}\big{(}86s_{1}^{3}-3s_{1}^{2}s_{2}-3s_{1}s_{2}^{2% }+86s_{2}^{3}\big{)}+2(s_{1}-s_{2})^{4}\big{(}9s_{1}^{2}-7s_{1}s_{2}+9s_{2}^{2% }\big{)}\bigg{]}\bigg{\}}+\frac{\pi^{2}s^{2}}{180\sigma^{3}}\bigg{\{}n_{F}% \bigg{[}43s^{6} \displaystyle-272s^{5}(s_{1}+s_{2})+s^{4}\big{(}715s_{1}^{2}+928s_{1}s_{2}+715% s_{2}^{2}\big{)}-20s^{3}\big{(}50s_{1}^{3}+39s_{1}^{2}s_{2}+39s_{1}s_{2}^{2}+5% 0s_{2}^{3}\big{)}+s^{2}\big{(}785s_{1}^{4}-508s_{1}^{3}s_{2}-114s_{1}^{2}s_{2}% ^{2}-508s_{1}s_{2}^{3}+785s_{2}^{4}\big{)} \displaystyle-4s\big{(}82s_{1}^{5}-251s_{1}^{4}s_{2}+109s_{1}^{3}s_{2}^{2}+109% s_{1}^{2}s_{2}^{3}-251s_{1}s_{2}^{4}+82s_{2}^{5}\big{)}+3(s_{1}-s_{2})^{2}\big% {(}19s_{1}^{4}-86s_{1}^{3}s_{2}+14s_{1}^{2}s_{2}^{2}-86s_{1}s_{2}^{3}+19s_{2}^% {4}\big{)}\bigg{]}+2n_{G}\bigg{[}23s^{6} \displaystyle-112s^{5}(s_{1}+s_{2})+s^{4}\big{(}215s_{1}^{2}+128s_{1}s_{2}+215% s_{2}^{2}\big{)}-40s^{3}\big{(}5s_{1}^{3}-6s_{1}^{2}s_{2}-6s_{1}s_{2}^{2}+5s_{% 2}^{3}\big{)}+s^{2}\big{(}85s_{1}^{4}-368s_{1}^{3}s_{2}+6s_{1}^{2}s_{2}^{2}-36% 8s_{1}s_{2}^{3}+85s_{2}^{4}\big{)} \displaystyle-8s\big{(}s_{1}^{5}-8s_{1}^{4}s_{2}+82s_{1}^{3}s_{2}^{2}+82s_{1}^% {2}s_{2}^{3}-8s_{1}s_{2}^{4}+s_{2}^{5}\big{)}-3(s_{1}-s_{2})^{2}\big{(}s_{1}^{% 4}-14s_{1}^{3}s_{2}-94s_{1}^{2}s_{2}^{2}-14s_{1}s_{2}^{3}+s_{2}^{4}\big{)}% \bigg{]}+4n_{S}\bigg{[}2s^{6}-13s^{5}(s_{1}+s_{2})+s^{4}\big{(}35s_{1}^{2} \displaystyle+47s_{1}s_{2}+35s_{2}^{2}\big{)}-5s^{3}\big{(}10s_{1}^{3}+9s_{1}^% {2}s_{2}+9s_{1}s_{2}^{2}+10s_{2}^{3}\big{)}+s^{2}\big{(}40s_{1}^{4}-17s_{1}^{3% }s_{2}-36s_{1}^{2}s_{2}^{2}-17s_{1}s_{2}^{3}+40s_{2}^{4}\big{)}+s\big{(}-17s_{% 1}^{5}+46s_{1}^{4}s_{2}+s_{1}^{3}s_{2}^{2}+s_{1}^{2}s_{2}^{3} \displaystyle+46s_{1}s_{2}^{4}-17s_{2}^{5}\big{)}+3(s_{1}-s_{2})^{6}\bigg{]}% \bigg{\}}\bar{B}_{0}(s)+\frac{\pi^{2}s_{1}^{2}}{180\sigma^{3}}\bigg{\{}n_{F}% \bigg{[}57s^{6}-4s^{5}(82s_{1}+93s_{2})+s^{4}\big{(}785s_{1}^{2}+1004s_{1}s_{2% }+615s_{2}^{2}\big{)}-4s^{3}\big{(}250s_{1}^{3}+127s_{1}^{2}s_{2} \displaystyle+109s_{1}s_{2}^{2}+150s_{2}^{3}\big{)}+s^{2}\big{(}715s_{1}^{4}-7% 80s_{1}^{3}s_{2}-114s_{1}^{2}s_{2}^{2}-436s_{1}s_{2}^{3}+615s_{2}^{4}\big{)}-4% s(s_{1}-s_{2})^{3}\big{(}68s_{1}^{2}-28s_{1}s_{2}-93s_{2}^{2}\big{)}+(s_{1}-s_% {2})^{5}(43s_{1}-57s_{2})\bigg{]} \displaystyle-2n_{G}\bigg{[}3s^{6}+8s^{5}(s_{1}-6s_{2})-s^{4}\big{(}85s_{1}^{2% }+64s_{1}s_{2}+195s_{2}^{2}\big{)}+8s^{3}\big{(}25s_{1}^{3}+46s_{1}^{2}s_{2}+8% 2s_{1}s_{2}^{2}+60s_{2}^{3}\big{)}-s^{2}\big{(}215s_{1}^{4}+240s_{1}^{3}s_{2}+% 6s_{1}^{2}s_{2}^{2}-656s_{1}s_{2}^{3} \displaystyle+195s_{2}^{4}\big{)}+16s(s_{1}-s_{2})^{3}\big{(}7s_{1}^{2}+13s_{1% }s_{2}+3s_{2}^{2}\big{)}-(s_{1}-s_{2})^{5}(23s_{1}+3s_{2})\big{)}+4n_{S}\bigg{% [}3s^{6}-s^{5}(17s_{1}+18s_{2})+s^{4}\big{(}40s_{1}^{2}+46s_{1}s_{2}+45s_{2}^{% 2}\big{)}+s^{3}\big{(}-50s_{1}^{3} \displaystyle-17s_{1}^{2}s_{2}+s_{1}s_{2}^{2}-60s_{2}^{3}\big{)}+s^{2}\big{(}3% 5s_{1}^{4}-45s_{1}^{3}s_{2}-36s_{1}^{2}s_{2}^{2}+s_{1}s_{2}^{3}+45s_{2}^{4}% \big{)}-s(s_{1}-s_{2})^{3}\big{(}13s_{1}^{2}-8s_{1}s_{2}-18s_{2}^{2}\big{)}+(s% _{1}-s_{2})^{5}(2s_{1}-3s_{2})\bigg{]}\bigg{\}}\bar{B}_{0}(s_{1}) \displaystyle+\frac{\pi^{2}s_{2}^{2}}{180\sigma^{3}}\bigg{\{}n_{F}\bigg{[}57s^% {6}-4s^{5}(93s_{1}+82s_{2})+s^{4}\big{(}615s_{1}^{2}+1004s_{1}s_{2}+785s_{2}^{% 2}\big{)}-4s^{3}\big{(}150s_{1}^{3}+109s_{1}^{2}s_{2}+127s_{1}s_{2}^{2}+250s_{% 2}^{3}\big{)}+s^{2}\big{(}615s_{1}^{4} \displaystyle-436s_{1}^{3}s_{2}-114s_{1}^{2}s_{2}^{2}-780s_{1}s_{2}^{3}+715s_{% 2}^{4}\big{)}-4s(s_{1}-s_{2})^{3}\big{(}93s_{1}^{2}+28s_{1}s_{2}-68s_{2}^{2}% \big{)}+(s_{1}-s_{2})^{5}(57s_{1}-43s_{2})\bigg{]}-2n_{G}\bigg{[}3s^{6}+8s^{5}% (s_{2}-6s_{1}) \displaystyle-s^{4}\big{(}195s_{1}^{2}+64s_{1}s_{2}+85s_{2}^{2}\big{)}+8s^{3}% \big{(}60s_{1}^{3}+82s_{1}^{2}s_{2}+46s_{1}s_{2}^{2}+25s_{2}^{3}\big{)}-s^{2}% \big{(}195s_{1}^{4}-656s_{1}^{3}s_{2}+6s_{1}^{2}s_{2}^{2}+240s_{1}s_{2}^{3}+21% 5s_{2}^{4}\big{)}-16s(s_{1}-s_{2})^{3} \displaystyle\times\big{(}3s_{1}^{2}+13s_{1}s_{2}+7s_{2}^{2}\big{)}+(s_{1}-s_{% 2})^{5}(3s_{1}+23s_{2})\bigg{]}+4n_{S}\bigg{[}3s^{6}-s^{5}(18s_{1}+17s_{2})+s^% {4}\big{(}45s_{1}^{2}+46s_{1}s_{2}+40s_{2}^{2}\big{)}+s^{3}\big{(}-60s_{1}^{3}% +s_{1}^{2}s_{2}-17s_{1}s_{2}^{2} \displaystyle-50s_{2}^{3}\big{)}+s^{2}\big{(}45s_{1}^{4}+s_{1}^{3}s_{2}-36s_{1% }^{2}s_{2}^{2}-45s_{1}s_{2}^{3}+35s_{2}^{4}\big{)}-s(s_{1}-s_{2})^{3}\big{(}18% s_{1}^{2}+8s_{1}s_{2}-13s_{2}^{2}\big{)}+(s_{1}-s_{2})^{5}(3s_{1}-2s_{2})\bigg% {]}\bigg{\}}\bar{B}_{0}(s_{2}) \displaystyle+\frac{\pi^{2}s^{2}s_{1}^{2}s_{2}^{2}}{3\sigma^{3}}\bigg{\{}n_{F}% \bigg{[}3s^{3}-3s^{2}(s_{1}+s_{2})-s\big{(}3s_{1}^{2}+2s_{1}s_{2}+3s_{2}^{2}% \big{)}+3(s_{1}-s_{2})^{2}(s_{1}+s_{2})\bigg{]}+n_{G}\bigg{[}-6s^{3}+6s^{2}(s_% {1}+s_{2})+s\big{(}6s_{1}^{2}+28s_{1}s_{2}+6s_{2}^{2}\big{)} \displaystyle-6(s_{1}-s_{2})^{2}(s_{1}+s_{2})\bigg{]}-4n_{S}ss_{1}s_{2}\bigg{% \}}C_{0}(s,s_{1},s_{2})-\frac{8\pi^{2}}{720}\,(s^{2}+s_{1}^{2}+s_{2}^{2})\big{% (}n_{S}+11n_{F}+62n_{G}\big{)}. (G.5)

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