]>NUPHB14503S0550-3213(18)30332-810.1016/j.nuclphysb.2018.11.016The Author(s)High Energy Physics – TheoryFig. 1One-loop diagrams for the TJJ in QED (continuous fermion lines). A similar set of diagrams describes the scalar QED case (dashed lines), by replacing the internal fermion with a charged scalar.Fig. 1Fig. 2The vertices in QED and scalar QED.Fig. 2Fig. 3(a): Dispersive description of the singularity of the spectral density ρ(s) as a spacetime process. (b): The exchange of a pole as the origin of the conformal anomaly in the TJJ viewed in perturbation theory.Fig. 3Table 1Basis of 13 fourth rank tensors satisfying the vector current conservation on the external lines with momenta p and q.Table 1itiμναβ(p,q)

1(k2gμν−kμkν)uαβ(p.q)

2(k2gμν−kμkν)wαβ(p.q)

3(p2gμν−4pμpν)uαβ(p.q)

4(p2gμν−4pμpν)wαβ(p.q)

5(q2gμν−4qμqν)uαβ(p.q)

6(q2gμν−4qμqν)wαβ(p.q)

7[p⋅qgμν−2(qμpν+pμqν)]uαβ(p.q)

8[p⋅qgμν−2(qμpν+pμqν)]wαβ(p.q)

9(p⋅qpα−p2qα)[pβ(qμpν+pμqν)−p⋅q(gβνpμ+gβμpν)]

10(p⋅qqβ−q2pβ)[qα(qμpν+pμqν)−p⋅q(gανqμ+gαμqν)]

11(p⋅qpα−p2qα)[2qβqμqν−q2(gβνqμ+gβμqν)]

12(p ⋅ q qβ − q2pβ) [2 pαpμpν − p2(gανpμ + gαμpν)]

13(pμqν + pνqμ)gαβ + p ⋅ q (gανgβμ + gαμgβν)−gμνuαβ − (gβνpμ + gβμpν)qα − (gανqμ + gαμqν)pβ

Exact correlators from conformal Ward identities in momentum space and the perturbative TJJ vertexClaudioCorianò⁎claudio.coriano@le.infn.itMatteo MariaMagliomatteomaria.maglio@le.infn.itDipartimento di Matematica e Fisica “Ennio De Giorgi”, Università del Salento and INFN Lecce, Via Arnesano, 73100 Lecce, ItalyDipartimento di Matematica e Fisica “Ennio De Giorgi”Università del SalentoINFN LecceVia ArnesanoLecce73100Italy⁎Corresponding author.Editor: Leonardo RastelliAbstractWe present a general study of 3-point functions of conformal field theory in momentum space, following a reconstruction method for tensor correlators, based on the solution of the conformal Ward identities (CWI's), introduced in recent works by Bzowski, McFadden and Skenderis (BMS). We investigate and detail the structure of the CWI's, their non-perturbative solutions and the transition to momentum space, comparing them to perturbation theory by taking QED as an example. We then proceed with an analysis of the TJJ correlator, presenting independent and detailed re-derivations of the conformal equations in the reconstruction method of BMS, originally formulated using a minimal tensor basis in the transverse traceless sector. A careful comparison with a second basis introduced in previous studies shows that this correlator is affected by one anomaly pole in the graviton (T) line, induced by renormalization. The result shows that the origin of the anomaly, in this correlator, should be necessarily attributed to the exchange of a massless effective degree of freedom. Our results are then exemplified in massless QED at one-loop in d-dimensions, expressed in terms of perturbative master integrals. An independent analysis of the Fuchsian character of the solutions, which bypasses the 3K integrals, is also presented. We show that the combination of field theories at one-loop – with a specific field content of degenerate massless scalar and fermions – is sufficient to generate the complete non-perturbative solution, in agreement with a previous study in coordinate space. The result shows that free conformal field theories, in specific dimensions, arrested at one-loop, reproduce the general result for the TJJ. Analytical checks of this correspondence are presented in d=3,4 and 5 spacetime dimensions. This implies that the generalized 3K integrals of the BMS solution can be expressed in terms of the two single master integrals B0 and C0 of 2- and 3-point functions, with significant simplifications.1IntroductionThe analysis of multi-point correlation functions in conformal field theory (CFT) is of outmost importance in high energy physics and in string theory, where exact results for lower (2- and 3-)point functions are combined with the operator product expansion (OPE) in order to characterize the structure of correlators of higher orders. This is the key motivation for a bootstrap program in d=4 spacetime dimensions.The enlarged SO(2,4) symmetry of CFT's – respect to Poincaré invariance – has been essential for establishing the form of some of their correlation functions. For 3-point functions, the solution of the conformal constraints in coordinate space allows to determine such correlators only up to few constants [1,2], which can then be fixed within a specific realization of a theory. In the case of a Lagrangian realization of a given CFT, such constants are expressed in terms of its (massless) field content (number of scalars, vectors, fermions), according to rather simple algebraic relations.Except for perturbative studies performed at Lagrangian level, such as in the case of the N=4 super Yang–Mills theory, which reach considerably high orders in the gauge coupling expansion, most of these analyses are performed in coordinate space, with no reference to any specific Lagrangian.There are obvious reasons for this. The first is that the inclusion of the conformal constraints is more straightforward to obtain in coordinate space, compared to momentum space. The second is that the operator product expansion (OPE) in momentum space is difficult to perform, especially for correlators of higher orders (≥3), in the Minkowski region. However, there are also some advantages which are typical of a momentum space analysis, and these are related to the availability of dimensional regularization (DR), at least at perturbative level, and to the technology of master integrals, which has allowed to compute large classes of multiloop amplitudes.Another advantage has to do with the identification of the conformal anomaly [3], which can be automatically extracted in DR (in d spacetime dimensions), being proportional to the 1/(d−4) singularity of the corresponding correlators. In coordinate space, instead, the anomaly contributions has to be added by hand by the inclusion of an inhomogeneous local term (i.e. by pinching all of its external coordinates), whose structure has to be inferred indirectly [1].Finally, a crucial issue concerns the physical character of the anomaly, which does not find any simple particle interpretation in position space, while it is clearly associated to the appearance of an anomaly pole in momentum space [4–6] in an uncontracted anomaly vertex. One finds, by a perturbative one-loop analysis of any anomalous correlator, that the anomaly is always associated with such massless exchanges in the corresponding diagram. It is therefore possible to identify them as effective degrees of freedom induced by the anomaly, present in the 1PI (one-particle irreducible) effective action. The physical significance of such contributions has been stressed in several previous works [4,6,7] along the years. They have recently discussed in condensed matter theory in the context of topological insulators and of Weyl semimetals [8,9].One of the goals of our work is to compare and extend previous perturbative analysis of the TJJ correlator with more recent ones based on the solution of the conformal Ward identities (CWI's) in momentum space [11–14]. This correlator is the simplest one describing the coupling of gravity to ordinary matter in QED and it has been investigated in perturbation theory from several directions [15–19].1.1Direct Fourier transform and the reconstruction programIn principle, one can move from coordinate space to position space in a CFT by a Fourier transform. This was the approach of [20] for 3-point functions, which can be explicitly worked out by introducing a regulator (ω) for the transform very much alike DR. The regulator serves as an intermediate step since some of the components of the correlators in position space are apparently non-transformable. It has been shown that 1/ω poles generated by the transform cancel in all the correlators analyzed, giving a complete expression for these in momentum space. The result is expressed in terms of ordinary and logarithmic master integrals of Feynman type, for which, in the latter case, it is possible to derive recursion relations as for ordinary ones [20]. The advantage of such approach is of being straightforward and algorithmic. It may be essential and probably the only manageable way to re-express the bootstrap program of CFT's in momentum space beyond 3- and 4-point functions, from the original coordinate space analysis. Consistency with the analysis presented in [20] implied rather directly that such logarithmic integrals had to be re-expressed in terms of ordinary Feynman integrals. In fact, it was shown in the same study that the TJJ correlator was entirely reproduced by a free field theory in coordinate space. Our analysis in momentum space is in complete agreement with this former result.1.2ReconstructionAn alternative method has been developed more recently, based on the direct solution of the conformal Ward identities in momentum space. The method has been proposed in [14] and [24] for scalar 3-point functions and extensively generalized to tensor correlators in [14].Several issues related to the renormalization of the solutions of the conformal Ward identities have been investigated in [12,13], adopting the formalism of the 3K integrals (i.e. parametric integrals of 3 Bessel functions). Several analyses in momentum space, for specific applications, have been worked out [10,23], but the generality of the approach is clearly a significant feature of [14], which reconstructs a tensor correlator starting from its transverse/traceless components and using the conservation/trace Ward identities (local terms). The latter are reconstructed from lower point functions.The result is expressed in terms of two sets of primary and secondary conformal Ward identities (CWI's), the first involving the form factors of the transverse/traceless contributions, which are parametrized on a symmetry basis, the second emerging from CWI's of lower point functions. For 3-point functions, the secondary CWI's involve conservation, trace and special WI's. In all the cases, the reconstructed solutions for 3-point functions can be given in terms of generalized hypergeometrics of type F4, [20], also known as Appell's hypergeometric function of two variables (F4), related to 3-K integrals [14].1.2.1The anomaly pole of the TJJOne of the results of our analysis will be to show how such contributions originate from the process of renormalization, taking as an example the case of the TJJ, filling in the intermediate steps of the discussion presented in [22]. We follow the general (BMS) approach introduced in [11] for the solution of the conformal constraints, which we detail in several of its parts, not offered in [11]. It has been compelling to proceed with an independent re-derivation of all the lengthy equations. The method can be directly generalized to higher point functions and implemented algorithmically, as we are going to show in a separate work. When coming to discuss the momentum space approach in CFT, there are several gaps in the literature, which are of methodological nature and need to be addressed. These concerns the correct form of the differential equations, the treatment of the derivatives of the Dirac δ's induced by momentum conservation, violations of the Leibnitz rule for the special conformal transformations, or the choice of the Lorentz (spin) singlet operator in the action of the conformal group on a specific correlator. These are points that we will address systematically. We will illustrate how to merge the results of the BMS approach on the structure of the minimal set of (4) form factors (the A-basis), solutions of the CWI's for the TJJ correlator, with a basis of 13 ones (the F-basis) defined in previous perturbative studies. We will show how to extract from the F-basis 4 combinations of the 13 and we will verify that they respect the scalar equations identified within the BMS approach.The use of this second basis is essential in order to prove that the WI's and the renormalization procedure for this correlator, imply that the anomaly can be attributed to the appearance of an anomaly pole in a single tensor structure of nonzero trace.1.3Our workAs we have just mentioned, one of the goals of this work, in a first part, is to present a systematic approach to the analysis of the CWI's in momentum space, closing a gap in the literature. The transition to momentum space raises the issue of how to include momentum conservation (i.e. translational invariance in coordinate space) in the presence of the dilatation and the special conformal generators. We will be dealing, in particular, with a rigorous treatment of such contributions which show up after a Fourier transform of the conformal generators to momentum space.We are going to investigate in detail the role of these contributions relying on the theory of tempered distributions. In particular, the discussion of these terms will be performed using a Gaussian basis which converges – in a distributional sense – to a covariant δ function in D=4 and allows to define a formal calculus for such distributions.Such contributions do not cancel, but lead to specific forms of the conformal generators in momentum space which are, however, in agreement with those presented in [14,21,24]. We define operational methods for the treatment of the covariant derivatives of δ functions in a consistent way, which may find application also beyond the scope of the current treatment, being quite general.In a second part we move to discuss scalar and tensor correlators and the solutions of the CWI's. We elaborate, in particular, one the apparent violation of the Leibnitz rule for the special conformal (SC) generator (Kκ), which emerges whenever we impose momentum conservation and eliminate one of the momenta, and the symmetric action of this operator, at an intermediate stage, is not evident. In position space this corresponds to choosing one coordinate to be zero, and treating the corresponding operator in a given correlation function, as spin singlet. The derivation of the constraints on the form factors is performed, in our case, by using Lorentz Ward identities, on which we elaborate in detail, confirming the results of [14]. We show how different choices for the singlet operator leads to an equivalent set of conformal equations. We then illustrate how to derive the solutions of the various form factors using some properties of the hypergeometric equations, bypassing the 3K integrals, showing that the Fuchsian indices of all the equations remain the same for all the solutions.Such analysis is followed by a perturbative study of TJJ correlator in the transverse traceless basis both in QED and in scalar QED, deriving the associated anomalous conformal Ward identities from this perspective.In the final part of our work we show how the perturbative solutions for the Ai, which are given in an appendix, reproduce the exact BMS result in a simplified way. We use the cases of d=3 and d=5 to show the exact correspondence between the two. This correspondence is studied by fixing an appropriate normalization of the photon two-point functions, on which we elaborate. This shows that the choice of different perturbative sectors (scalar, fermion) in both cases are sufficient to reproduce the entire nonperturbative result. This implies that only arbitrary constant in the nonperturbative solution, expressed in terms of the 3K integrals, has to simplify and be expressible in terms of simple integrals B0 and C0, the scalar 2- and 3-point functions. In our conclusions we briefly comment on the possible origin of such simplifications.2Special conformal Ward identities in the operatorial approachIn this section, to make our treatment self-contained, we briefly illustrate the operatorial derivation of the CWI's for correlators involving 3-point functions of stress energy tensors.An infinitesimal transformation(2.1)xμ(x)→x′μ(x)=xμ+vμ(x) is classified as an isometry if it leaves the metric gμν(x) invariant in form. If we denote with gμν′(x′) the new metric in the coordinate system x′, then an isometry is such that(2.2)gμν′(x′)=gμν(x′). This condition can be inserted into the ordinary covariant transformation rule for gμν(x) to give(2.3)gμν′(x′)=∂xρ∂x′μ∂xσ∂x′νgρσ(x)=gμν(x′) from which one derives the Killing equation for the metric(2.4)vα∂αgμν+gμσ∂νvσ+gσν∂μvσ=0. For a conformal transformation the metric condition (2.2) is replaced by the condition(2.5)gμν′(x′)=Ω−2gμν(x′) generating the conformal Killing equation (with Ω(x)=1−σ(x))(2.6)vα∂αgμν+gμσ∂νvσ+gσν∂μvσ=2σgμν. In the flat spacetime limit this becomes(2.7)∂μvν+∂νvμ=2σημν,σ=1d∂⋅v. From now on we switch to the Euclidean case, neglecting the index positions. Using the fact that every conformal transformation can be written as a local rotation matrix of the form(2.8)Rαμ=Ω∂x′μ∂xα we can first expand generically R around the identity as(2.9)R=1+[ϵ]+… with an antisymmetric matrix [ϵ], which we can re-express in terms of antisymmetric parameters (τρσ) and 1/2d(d−1) generators Σρσ of SO(d) as(2.10)[ϵ]μα=12τρσ(Σρσ)μα(Σρσ)μα=δρμδσα−δραδσμ from which, using also (2.8) we derive a constraint between the parameters of the conformal transformation (v) and the parameters τμα of R(2.11)Rμα=δμα+τμα=δμα+12∂[αvμ] with ∂[αvμ]≡∂αvμ−∂μvα.Denoting with ΔA the scaling dimensions of a vector field Aμ(x)′, its variation under a conformal transformation can be expressed via R in the form(2.12)A′μ(x′)=ΩΔARμαAα(x)=(1−σ+…)ΔA(δμα+12∂[αvμ]+…)Aα(x) from which one can easily deduce that(2.13)δAμ(x)≡A′μ(x)−Aμ(x)=−(v⋅∂+ΔAσ)Aμ(x)+12∂[αvμ]Aα(x), which is defined to be the Lie derivative of Aμ in the v direction, modulo a sign(2.14)LvAμ(x)≡−δAμ(x). As an example, in the case of a generic rank-2 tensor field (ϕIK) of scaling dimension Δϕ, transforming according to a representation DJI(R) of the rotation group SO(d), (2.12) takes the form(2.15)ϕ′IK(x′)=ΩΔϕDI′I(R)DK′K(R)ϕI′K′(x). In the case of the stress energy tensor (D(R)=R), with scaling (mass) dimension ΔT (ΔT=d) the analogue of (2.12) is(2.16)T′μν(x′)=ΩΔTRαμRνβTαβ(x)=(1−ΔTσ+…)(δμα+12∂[αvμ]+…)(δμα+12∂[αvμ]+…)Tαβ(x) where ∂[αvμ]≡∂αvμ−∂μvα. One gets(2.17)δTμν(x)=−ΔTσTμν−v⋅∂Tμν(x)+12∂[αvμ]Tαν+12∂[νvα]Tμα.For a special conformal transformation (SCT) one chooses(2.18)vμ(x)=bμx2−2xμb⋅x with a generic parameter bμ and σ=−2b⋅x (from 2.7) to obtain(2.19)δTμν(x)=−(bαx2−2xαb⋅x)∂αTμν(x)−ΔTσTμν(x)+2(bμxα−bαxμ)Tαν+2(bνxα−bαxν)Tμα(x).It is sufficient to differentiate this expression respect to bκ in order to derive the form of the SCT Kκ on T in its finite form(2.20)KκTμν(x)≡δκTμν(x)=∂∂bκ(δTμν)=−(x2∂κ−2xκx⋅∂)Tμν(x)+2ΔTxκTμν(x)+2(δμκxα−δακxμ)Tαν(x)+2(δκνxα−δακxν)Tμα. The approach can be generalized to correlators built out of several operators. In the case of a TJJ correlator,(2.21)Γμναβ(x1,x2,x3)=〈Tμν(x1)Jα(x2)Jβ(x3)〉 with a vector current of dimension ΔJ, the CWI's take the explicit form(2.22)KκΓμναβ(x1,x2,x3)=∑i=13Kiscalarκ(xi)Γμναβ(x1,x2,x3)+2(δμκx1ρ−δρκx1μ)Γρναβ+2(δνκx1ρ−δρκx1ν)Γμραβ2(δακx2ρ−δρκx2α)Γμνρβ+2(δβκx3ρ−δρκx3β)Γμναρ=0, where(2.23)Kiscalarκ=−xi2∂∂xκ+2xiκxiτ∂∂xiτ+2Δixiκ is the scalar part of the special conformal operator acting on the ith coordinate and Δi≡(ΔT,ΔJ,ΔJ) are the scaling dimensions of the operators in the correlation function.2.1Constraints from translational symmetry and the Leibniz ruleOne of the main issues, when moving to momentum space, is to include the constraint from translational symmetry on a tensor correlator. The inclusion of this constraint at the beginning, for a tensor 3-point function of the form(2.24)〈H1μ1ν1(x1)H2μ2ν2(x2)H3μ3ν3(x3)〉, with each of the Hi's of scaling dimensions ΔiH, reduces it to the form(2.25)〈H1μ1ν1(x13)H2μ2ν2(x23)H3μ3ν3(0)〉, and the action of Kκ, the special conformal generator, on (2.24) and (2.25) will obviously change. The transition to momentum space in the two cases above takes to two different forms of the special CWI. The first form will be symmetric in momentum space, but at the cost of generating derivatives of the delta-function, which enforce conservation of the total momentum, while the second one will be asymmetric, treating one of the momenta as dependent from the other two. The final result for the scalar equations of the corresponding form factors will obviously be symmetric respect the three momenta.In particular, the Lorentz (spin) generator, in the case of (2.25), will act only the indices of H1 and H2, but not on those of H3, although the differentiation respect to the 4-momentum p3 will be performed implicitly by a chain rule in this second case, once we move to momentum space.We are going to illustrate this point in detail.Identifying KiκHiμν with the expression (2.20) (with ΔT→ΔiH), then the action of the special conformal transformation on (2.24) will take the forms(2.26)Kκ〈H1μ1ν1(x1)H2μ2ν2(x2)H3μ3ν3(x3)〉=∑i=13Kiκ〈H1μ1ν1(x1)H2μ2ν2(x2)H3μ3ν3(x3)〉=e−iPx3Kκ′〈H1μ1ν1(x13)H2μ2ν2(x23)H3μ3ν3(0)〉 with P the total translation operator (P=P1+P2+P3) and(2.27)Kκ′=eiPx3Kκe−iPx3. Using the relations of the conformal algebra[Kκ,Pν]=2i(ηκνD+Mκν),[D,Pμ]=−iPμ[Mκν,Pμ]=−i(ηκμPν−ημνPκ), and expanding (2.27) we obtain the relation(2.28)Kκ′=Kκ+ix3μ[Pμ,Kκ]+i22x3μx3ν[Pμ,[Pν,Kκ]]+…=Kκ+2x3κD+2x3μMκμ−2x3κx3μPμ+x32Pκ, since the commutator of higher order vanish.The explicit form of the operators (dilatation, Lorentz and special conformal) D,Mμν,Kκ is Kκ=K1κ+K2κ+K3κ, D=D1+D2+D3, Mμν=M1μν+M2μν+M3μν and(2.29)Miμν=Liμν+Σiμν,Ll=i(xlμ∂∂xlν−xlν∂∂xlμ), split into angular momentum (L) and spin (Σ). We illustrate this crucial point in some detail, since it shows how the action of the Lorenz generators on the field at x3 vanishes. We get (using pˆlκ≡i∂/∂xlκ)(2.30)Kκ′=x12pˆ1κ−2x1κx1νpˆ1ν−2ix1κΔ1−2x1νΣκν1+x22pˆ2κ−2x2κx2νpˆ2ν−2ix2κΔ2−2x2νΣκν2+x32pˆ3κ−2x3κx3νpˆ3ν−2ix3κΔ3−2x3νΣ3κν+2ix3κ(Δ1+Δ2+Δ3)+2x3κ(x1νpˆ1ν+x2νpˆ2ν+x3νpˆ3ν)+2x3ν(Σ1κν+Σ2κν+Σ3κν)+2x3ν(x1κpˆ1ν−x1νpˆ1κ+x2κpˆ2ν−x2νpˆ2κ+x3κpˆ3ν−x3νpˆ3κ)−2x3κx3μ(pˆ1μ+pˆ2μ+pˆ3μ)+x32(pˆ1κ+pˆ2κ+pˆ3κ) which shows the cancellation of the contribution from the generator M3 since(2.31)Kκ′=(x1−x3)2pˆ1κ−2(x1κ−x3κ)(x1ν−x3ν)p1ˆν−2i(x1κ−x3κ)Δ1−2(x1ν−x3ν)Σ1κν+(x2−x3)2pˆ2κ−2(x2κ−x3κ)(x2ν−x3ν)pˆ2ν−2i(x2κ−x3κ)Δ2−2(x2ν−x3ν)Σ2κν. Notice that both Σ1 and Σ2 denote the two spin matrices, which act only on H1 and H2. In other words, in coordinate space the choice x3=0 implies that H3 behaves as a Lorentz singlet respect to the spin part. The result can be rewritten in the compact form(2.32)Kκ′=K13κ+K23κ, where the action of K13κ on a rank-2 tensor Hμν, for instance, is given by(2.33)KκHμν(x13)=Kscalarκ(x13)Hμν+2(δμκx13ρ−δρκx13μ)Hρν+2(δνκx13ρ−δρκx13ν)Hμρ, with Kscalar(x13) being given as in (2.23) with xi→x13 and Δi→ΔH, and we obtain the equation(2.34)0=Kκ〈H1μ1ν1(x1)H2μ2ν2(x2)H3μ3ν3(x3)〉=e−iPx3(K13κ+K23κ)〈H1μ1ν1(x13)H2μ2ν2(x23)H3μ3ν3(0)〉. Notice that the solution of this equation can be obtained by solving the reduced equation(2.35)(K13κ+K23κ)〈H1μ1ν1(x13)H2μ2ν2(x23)H3μ3ν3(0)〉=0 which is equivalent to finding the solution of (2.34) with x3=0 and acting afterwards with the translation operator e−iPx3 to restore the full dependence on the third coordinate. Setting(2.36)χμ1ν1μ2ν2μ3ν3(x12,x13)≡〈H1μ1ν1(x13)H2μ2ν2(x23)H3μ3ν3(0)〉, anticipating the discussion that will be presented for these equations in momentum space, the special CWI then can be cast into the form(2.37)∫d4p1d4p2e−i(p1x13+p2x23)(Kp1κ+Kp2κ)χ(p1,p2)=0 giving(2.38)(Kp1κ+Kp2κ)χ(p1,p2)=0. Notice that the previous form of χ(p1,p2), which is a function of the independent momenta p1 and p2, conjugate to x12 and x13, is the final form of the function, having re-expressed p3 in terms of p1 and p2. In a direct explicit computation, one has to act with the transforms of Kκ(x12) and Kκ(x13), that we denote as Kκ(p1) and Kκ(p2), on the transform of the initial correlator(2.39)(Kκ(p1)+Kκ(p2))〈H1μ1ν1(p1)H2μ2ν2(p2)H3(p¯3)〉==∫ddx13ddx23e−ip1⋅x12−ip2⋅x23(Kκ(x12)+Kκ(x13))×〈H1μ1ν1(x13)H2μ2ν2(x23)H3μ3ν3(0)〉 with p3→p¯3=−p1−p2 and the Leibniz rule is violated. The symmetry respect to the external invariants (p12,p22,p32) of the conformal generator is only reobtained at the end, after applying the chain rule for the differentiation of p3 respect to the two independent momenta.The final result is that in momentum space we can treat H3(p¯3) as a single particle operator, in the sense that the differentials in p1 and p2 will act separately on H1 and H2, but also on H3 implicitly, via a chain rule.At the same time, as clear from (2.31), the spin rotation matrix Σμν contained in the Lorentz generator Mμν will act only on H1 and H2, treating H3 as a Lorentz (spin) singlet. We will present on the sections below complete worked out examples of this action. Since we are free to set any of the 3 coordinates to zero, the intermediate steps of the computations of the CWI's will be completely different, and the choice of the Lorentz singlet operator can be dictated by convenience.The choice of the point x which will be set to zero (e.g. x3=0) is obviously arbitrary, but preferably should be suggested by the symmetry of the correlator. For instance, for correlators such as 〈T(x1)T(x2)T(x3)〉 and 〈T(x1)T(x2)O(x3)〉 setting x3=0 and removing momentum p3 in terms of p1 and p2 is the natural choice. In the 〈T(x1)J(x2)J(x3)〉 case it is convenient to set x1=0 and re-express the momentum p1 in terms of p2 and p3.3The conformal generators in momentum spaceIn this section we discuss two formulations of the dilatation and SCT's, with the goal of clarifying the treatment of the constraints coming from the conservation of the total momentum in a generic correlator. We will be using some condensed notations in order to shorten the expressions of the transforms in momentum space. We will try to avoid the proliferation of indices, whenever necessary, with the conventions(3.1)Φ(x_)≡〈ϕ1(x1)ϕ2(x2)…ϕn(xn)〉eipx_≡ei(p1x1+p2x2+…pnxn)dp_≡dp1dp2…dpnΦ(p_)≡Φ(p1,p2,…,pn). It will also be useful to introduce the total momentum P=∑j=1npj.The momentum constraint is enforced via a delta function δ(P) under integration. For instance, translational invariance of Φ(x_) gives(3.2)Φ(x_)=∫dp_δ(P)eipx_Φ(p1,p2,p3). In general, for an n-point function Φ(x1,x2,…,xn)=〈ϕ1(x1)ϕ2(x2)...ϕn(xn)〉, the condition of translation invariance(3.3)〈ϕ1(x1)ϕ2(x2),…,ϕn(xn)〉=〈ϕ1(x1+a)ϕ2(x2+a)…ϕn(xn+a)〉 generates the expression in momentum space of the form (3.2), from which we can remove one of the momenta, conventionally the last one, pn, which is replaced by its “on shell” version p‾n=−(p1+p2+…pn−1)(3.4)Φ(x1,x2,…,xn)=∫dp1dp2...dpn−1ei(p1x1+p2x2+...pn−1xn−1+p‾nxn)Φ(p1,p2,…,p‾n). We start by considering the dilatation WI.The condition of scale covariance for the fields ϕi of scale dimensions Δi (in mass units)(3.5)Φ(λx1,λx2,…,λxn)=λ−ΔΦ(x1,x2,…,xn),Δ=Δ1+Δ2+…Δn after setting λ=1+ϵ and Taylor expanding up to O(ϵ) gives the scaling relation(3.6)(Dn+Δ)Φ≡∑j=1n(xjα∂∂xjα+Δj)Φ(x1,x2,…,xn)=0, with(3.7)Dn=∑j=1nxjα∂∂xjα. The corresponding equation in momentum space can be obtained either by a Fourier transform of (3.6), which can give either symmetric or asymmetric expressions of the equations in the respective momenta pi or, more simply, exploiting directly (3.5). In the latter case, using the translational invariance of the correlator under the integral, by removing the δ-function constraint, one obtains(3.8)Φ(λx1,λx2,…,λxn)=∫ddp1ddp2…ddpn−1eiλ(p1x1+p2x2+...pn−1xn−1+p‾nxn)Φ(p1,p2,…,p‾n)=λ−Δ∫ddp1ddp2…ddpn−1ei(p1x1+p2x2+...pn−1xn−1+p‾nxn)Φ(p1,p2,…,p‾n). It is simply a matter of performing the change of variables pi=pi′/λ on the rhs of the equation above (first line) with dp1...dpn−1=(1/λ)d(n−1)ddp1′…ddpn−1′ to derive the relation(3.9)1λd(n−1)Φ(p1λ,p2λ,…,p‾nλ)=λ−ΔΦ(p1,p2,…,p‾n). Setting λ=1/s this generates the condition(3.10)s(n−1)d−ΔΦ(sp1,sp2,…,sp‾n)=Φ(p1,p2,…,p‾n) and with s∼1+ϵ, expanding at O(ϵ) we generate the equation(3.11)[∑j=1nΔj−(n−1)d−∑j=1n−1pjα∂∂pjα]Φ(p1,p2,…,p‾n)=0. It is straightforward to reobtain the same equation from the direct Fourier transform of (3.6) if we use translational invariance since(3.12)(Dn+Δ)Φ(x1,…,xn)==(Dn+Δ)∫ddp1…ddpnδ(P)eip1x1+…pnxnΦ(p1,…,pn)=(∑j=1nΔj+∑i=1n−1xin∂∂xin)∫ddp1…ddpn−1eip1x1n+…pn−1xn−1n×Φ(p1,…,pn−1,p¯n)=∫ddp1…ddpn−1(∑j=1nΔj+∑j=1n−1pj∂∂pj)eip1x1n+…pn−1xn−1n×Φ(p1,…,pn−1,p¯n). At this point we perform a partial integration n−1 times, moving the derivatives from the exponential to the correlator Φ to reobtain (3.11)(3.13)0=(Dn+Δ)Φ(x1,…,xn)=∫ddp1…ddpn−1(∑j=1nΔj−(n−1)d−∑j=1n−1pjα∂∂pjα)×Φ(p1,…,pn−1,p¯n)eip1x1n+…ipn−1xn−1n. A rigorous way to reobtain this result is to consider directly the conformal algebra for the dilatation operator D=(iDn+Δ) and use the commutation relations. For our purpose we can consider a realization of Φ(x1,…xn) via some operators Oj(xj)(3.14)D〈O1(x1)…On(xn)〉≡∑j=1nD(xj)〈O1(x1)…On(xn)〉=[∑j=1nΔj+∑j=1nxjα∂∂xjα]〈O1(x1)…On(xn)〉.This n-point function is translationally invariant, so that we can shift the fields using the translation operator exp(iP⋅xn), with P the total translation operator P=∑j=1nPj(3.15)D〈O1(x1)…On(xn)〉=e−ixn⋅PD′〈O1(x1−xn)…On(xn−1−xn)On(0)〉=0 where(3.16)D′=eixnμ⋅PμDe−ixnμ⋅Pμ=∑k=0∞ikk!xnν1…xnνk[Pν1,[…[Pνk,D]…]]. Using the commutation relations of the conformal algebra, there are at most two non-vanishing terms in this sum. Evaluating the finite multiple commutators we get(3.17)D′=D+ixnν[Pν,D]=D−xnνPν and explicitly(3.18)D′=∑j=1n[Δj+xjν(Pj)ν]−xnν∑j=1n(Pj)ν=∑j=1n−1(xj−xn)ν(Pj)ν+∑j=1nΔj=∑j=1n−1(xj−xn)ν∂∂xjν+∑j=1nΔj. Notice that the solution of (3.15) can be obtained by solving the reduced equation(3.19)D′〈O1(x1−xn)…On(xn−1−xn)On(0)〉=0 and then acting with a total translation. In momentum space this relation generates the Ward identity(3.20)∫∏j=1nddxjeix1⋅p1+…ipn⋅xnD′〈O1(x1−xn)…On(xn−1−xn)On(0)〉=0, that is(3.21)∫ddp1…ddpnδ(d)(∑j=1npj)[∑j=1nΔj−(n−1)d−∑j=1n−1pjα∂∂pjα]×〈O1(p1)…On(pn−1)On(p¯n)〉=0, where p¯n=−∑j=1n−1pj.If we decided to work with symmetric expressions of the transform, the approach would be more cumbersome since it would involve δ′ (derivative) terms in the integrand. We are going to discuss this second approach both for the dilatation and for the special conformal transformations. It requires a brief digression on the use of some relations for the covariant δ-functions which we are going to formulate below and that will be essential in order to clarify the correct way to treat such contributions.3.1Delta calculus and symmetric GaussiansIt is possible to derive a formal calculus for the derivatives of δd(P) using as defining conditions that, for a generic function f(p) which is regular at Pμ=0 the rules(3.22)∫ddP∂αδd(P)f(P)=−∂αf(0) and(3.23)∫ddP∂β∂αδd(P)f(P)=∂β∂αf(0) hold. For instance one easily obtains formally(3.24)∂αδd(P)≡∂∂Pαδd(P)=−dP2δd(P)Pα, which can be checked using the following rule for symmetric integration in (3.22)(3.25)∫ddPδd(P)P2PαPβf(0)=1dηαβ, where f(0) is a constant. Similarly, one can show that(3.26)∫ddPPαδd(P)P2f(0)=0, which is consistent with the fact that the integral(3.27)∫ddP∂αδd(P)=0, has a zero boundary value. Notice that (3.26) can be extended to the more general form(3.28)∫ddPPαδd(P)P2f(P)=0, if the function f(P) is regular at P=0.To derive such relations on a rigorous basis, we need to introduce a suitable family of functions converging to the δd(P) in the distributional limit.We will be needing the relations(3.29)δd(P)=δ(P)δ(Ω)Pd−1C(θ1,...,θd−1),δ(Ω)≡∏l=1D−1δ(θl)C(θ1,...θd−1)=∏l=1d−2sinθld−l−1 between the cartesian and the polar coordinates versions of the delta function. Then clearly(3.30)∫ddPδd(P)=∫0∞dP∫0πdθ1∫0πdθ2…∫02πdθd−1δ(P)δ(Ωd)=1. It is easily shown that the integral of a vector n of unit norm, expressed in the same variables(3.31)nα(θ1,…,θd−1)=(cosθ1,cosθ2sinθ1,…,sinθd−1…sinθ1),0≤θi≤π,i=1,2,…d−2,0≤θd−1≤2π, vanishes(3.32)∫dΩnα(θ1,…,θd−1)=0,dΩ≡dθ1dθ2…dθd−1. These relations will be used to parametrize the tensor integrals over the (total) momentum Pμ of the correlators as nμ|P|, with |P| being the magnitude of P. Notice that respect to the d-dimensional angular integration measure dΩd, dΩ is stripped of the angular factors(3.33)dΩd≡dΩ∏l=1d−2sinθld−l−1.The vanishing of (3.32) is simply due to the symmetry of the angular integrations. This may not be obvious if we separate the angular from the radial parts of the δ function and performs the angular integration first, since the d-dimensional rotational symmetry is broken(3.34)∫dΩnα(θ1,…,θd−1)δ(Ω)=δα0, where the nonzero component surviving in (3.34) depends on the directions chosen for the polar axis, and the integration measure has been stripped off of the angular factors. Notice that only one component (n0), proportional to cosθd−1, is nonvanishing after the integration with δ(Ω), the remaining ones being zero. Therefore, the vanishing of (3.26) has to be shown by using a rotationally symmetric sequence of functions which avoid the formal manipulation in (3.32). For this purpose we use a sequence of normalized Gaussians(3.35)Gk(P)=1(2π)d/2kde−P22k2,∫ddPGk(P)=1 converging to δd(P) as k→0. We need to consider the distributional limit of(3.36)∫ddPPαP2Gk(P)=∫0∞Pd−2dPGk(P))∫nα(θ1,…,θd−1)dΩd=122πd/2kd/2+1/2Γ(d−12)∫nα(θ1,…,θd−1)dΩd which vanishes after angular integration since(3.37)Yα(n)≡∫nα(θ1,…,θd−1)dΩd=0 with the boundaries given in (3.31). Therefore, the correct angular average should be taken before the distributional limit of k→0, giving a vanishing result, thereby proving (3.26). The result for the rank-2 integral in (3.25) which has been justified above by covariance and symmetric integration, can also be obtained by a similar method. In this case we get(3.38)∫ddPPαPβP2Gk(P)=∫0∞Pd−1Gk(P)dPYαβ(n), where we have defined the angular part(3.39)Yαβ(n)≡∫nαnβ(θ1,…,θd−1)dΩd=1dVdδαβ,Vd=2πd/2Γ[d/2].Also in this case the integral (3.38) is factorized with(3.40)∫0∞dPPd−1Gk(P)=1Vd, which is independent of the k parameter of the distributional limit. Therefore(3.41)∫ddPPαPβP2Gk(P)=∫0∞Pd−1Gk(P)dPYαβ(n)=1dδαβ for any value of the parameter k. This takes directly to (3.25) if the parameter of the Gaussian family approaches k=0 in order to extract the value of a test function f(P) at P=0.One can expand on this result using the rules of the ordinary calculus formally, by taking multiple derivatives of δ(P) using (3.22), which imply that(3.42)∂β∂αδd(P)=∂β(−dP2δd(P)Pα)=d(d+2)(P2)2δd(P)PαPβ−dP2δd(P)δαβ. These correctly generate (3.23) using symmetric integration(3.43)∫ddPδd(P)(P2)2PαPβPρPσf(0)=1d(d+2)(δαβδρσ+δαρδβσ+δασδβρ).Another useful relation is(3.44)∑j=1npjα∂∂pjαδd(P)=∑j=1npjα∂∂Pαδd(P)=Pα∂αδd(P)=−dδd(P), using (3.24), which is of immediate derivation. We will be using the relations above to illustrate the elimination of one of the momenta from the differential equations which characterize the CWI's in momentum space.3.2The dilatation Ward identity with one less momentumWe can reobtain the results of the previous sections for the dilatation WI by using the calculus derived above. The dilatation Ward identity of (3.6) can indeed be written in a (p1,p2,…pn) symmetric form using(3.45)∑j=1n(xjα∂∂xjα+Δj)Φ(x1,x2,…,xn)=∑j=13(xjα∂∂xjα+Δj)∫ddp_eipx_δ(P)Φ(p_)=∫ddp_δd(P)eipx_(∑j=1nΔj−nd−∑j=1npjα∂∂pjα)Φ(p_)+δterm′, with(3.46)δterm′=−∫ddp_Pα∂αδd(P)eipx_Φ(p_)=d∫ddp_δd(P)eipx_Φ(p_), where we have used (3.44). Inserting (3.46) into (3.45) we obtain the symmetric expression of the scaling relation in momentum space(3.47)∑j=1n(xjα∂∂xjα+Δj)Φ(x1,x2,…,xn)=∫ddp_δd(P)eipx_(∑j=1nΔj−(n−1)d−∑j=1npjα∂∂pjα)Φ(p_). The expression given above depends only on n−1 momenta, since one of them can be eliminated. If we choose as independent ones p1,…pn−1 with (p1,…,pn)→(p1,…,pn−1,P), and define q=p1+p2…+pn−1, then the dependence on pn in (3.47) can be re-expressed in terms of the total momentum P and of the sum of the independent momenta q as(3.48)Φ(p_)=Φ(p1,…,pn−1,P)pnα=Pα−qα∑j=1npjα∂∂pjαΦ(p_)=∑j=1n−1pjα∂∂pjαΦ(p1,…pn−1,P)+(Pα−qα)∂∂PαΦ(p1,…pn−1,P). The last (nth) term in (3.47) is given by(3.49)σn≡∫ddp_δd(P)eipx_pnα∂pnαΦ(p_). Rewriting the exponential as eip⋅x_→ei(p1⋅x1n+…pn−1⋅xn−1n+iP⋅xn) and using the δd(P) to remove the P⋅xn term, σn takes the form(3.50)σn=∫ddp1ddp2…ddpn−1ddPδd(P)eip1x1n+ip2x2n+…ipn−1xn−1n××(P−q)α∂PαΦ(p1,…pn−1,P)=−∫ddp1ddp2…ddpn−1eip1x1n+ip2x2n+…ipn−1xn−1nqα×∫ddP∂∂PαΦ(p1,…pn−1,P)δd(P). Notice that in the expression above we have removed the δd(P)Pα∂∂PαΦ(p_) term, which after a partial integration becomes(3.51)∫ddPδd(P)Pα∂∂PαΦ(p1,…pn−1,P)eip1x1n+ip2x2n+…ipn−1xn−1n=∫ddPeip1x1n+ip2x2n+…ipn−1xn−1n(−d+Pα∂∂Pαδd(P))Φ(p1,…pn−1,P) and vanishes by (3.24). Similarly, we derive the vanishing relation(3.52)∫ddPPα∂∂PαΦ(p1,…pn−1,P)δd(P)=d∫ddPδd(P)PαP2Φ(p1,…pn−1,P)=d∫ddPδd(P)PαP2Φ(p1,…pn−1,0)(3.53)=0 which has been obtained as a result of (3.26) and (3.28). Therefore we find that σn=0, reobtaining the expected scaling equation in momentum space(3.54)(∑j=1nΔj−(n−1)d−∑j=1n−1pjα∂∂pjα)Φ(p1,…pn−1,p¯n)=0.3.3Special conformal WI's for scalar correlatorsWe now turn to the analysis of the special conformal transformations in momentum space. In this second case the δ′ terms cancel identically. Also in this case we discuss both the symmetric and the asymmetric forms of the equations, focusing our attention first on the scalar case. The Ward identity in the scalar case is given by(3.55)∑j=1n(−xj2∂∂xjκ+2xjκxjα∂∂xjα+2Δjxjκ)Φ(x1,x2,…,xn)=0 which in momentum space, using(3.56)xjα→−i∂∂pjα∂∂xjκ→ipjκ becomes(3.57)∑j=1n∫ddp_(pjκ∂2∂pjα∂pjκ−2pjα∂2∂pjα∂pjκ−2Δj∂∂pjκ)eip⋅x_δd(P)ϕ(p_)=0, where the action of the operator is only on the exponential. At this stage we integrate by parts, bringing the derivatives from the exponential to the correlator and on the Dirac δ function obtaining(3.58)∫ddp_eipx_KskΦ(p_)δd(P)+δterm′=0 in the notations of Eq. (3.1), where we have introduced the differential operator acting on a scalar correlator in a symmetric form(3.59)Ksk=∑j=1n(pjκ∂2∂pjα∂pjα+2(Δj−d)∂∂pjκ−2pjα∂2∂pjκ∂pjα).Some of the terms containing first and second derivatives of the Dirac delta function can be rearranged using also the intermediate relation(3.60)Ksκδd(P)=(Pk∂2∂Pα∂Pα−2Pα∂2∂Pα∂Pk+2(Δ−nd)∂∂Pk)δd(P)=2d(dn−d−Δ)Pkδd(P)P2=−2(dn−d−Δ)∂∂Pkδd(P),Δ=∑j=1nΔj, where we have repeatedly used (3.24) together with (3.42) and (3.44). Combining all the derivative terms, on the other hand, we obtain(3.61)δterm′=∫ddp_eip⋅x_[∂∂Pαδd(P)∑j=1n(pjα∂∂pjκ−pjκ∂∂pjα)Φ(p_)+2∂∂Pκδd(P)(∑j=1n(Δj−pjα∂∂pjα)−(n−1)d)Φ(p_)]. Notice that such terms vanish by using rotational invariance of the scalar correlator(3.62)∑j=13(pjα∂∂pjκ−pjκ∂∂pjα)Φ(p_)=0, as a consequence of the SO(4) symmetry(3.63)∑j=13Lμν(xj)〈ϕ(x1)ϕ(x2)ϕ(x3)〉=0, with(3.64)Lμν(x)=i(xμ∂ν−xν∂μ), and the symmetric scaling relation,(3.65)(∑j=1nΔj−∑jn−1pjα∂∂pjα−(n−1)d)Φ(p_)=0, where we have used (3.24). Using (3.60) and the vanishing of the δterm′ terms, the structure of the CWI on the correlator Φ(p) then takes the symmetric form(3.66)∑j=1n∫ddp_eip⋅x_(pjκ∂2∂pjα∂pjα−2pjα∂2∂pjα∂pjκ+2(Δj−d)∂∂pjκ)ϕ(p_)δd(P)=0. This symmetric expression is the starting point in order to proceed with the elimination of one of the momenta, say pn. Also in this case, one can proceed by following the same procedure used in the derivation of the dilatation identity, dropping the contribution coming from the dependent momentum pn, thereby obtaining the final form of the equation(3.67)∑j=1n−1(pjκ∂2∂pjα∂pjα+2(Δj−d)∂∂pjκ−2pjα∂2∂pjκ∂pjα)Φ(p1,…pn−1,p¯n)=0. Also in this case the differentiation respect to pn requires the chain rule. For a certain sequence of scalar single particle operators(3.68)Φ(p1,…pn−1,p¯n)=〈ϕ(p1)…ϕ(p¯n)〉 the Leibnitz rule is therefore violated. As we have already mentioned, the complete symmetry of the solution respect to the three momenta is however respected. We will now move to a discussion of the general structure of the method, focusing first on scalars and then on tensor correlators.4Reduction of the action of KscalarκIn the case of a scalar correlator all the anomalous conformal WI's can be re-expressed in scalar form by taking as independent momenta the magnitude pi=pi2 as the three independent variables. Defining F(p1,p2)=Φ(p1,p2,p3¯) and using the relation(4.1)p1α∂Fp1α+p2α∂Fp2α=p1∂Φ∂p1+p2∂Φ∂p2+p3∂Φ∂p3 the anomalous scale equation becomes(4.2)(Δ−2d−∑i=13pi∂∂pi)Φ(p1,p2,p¯3)=0. The relation above is derived using the chain rule(4.3)∂Φ∂piμ=piμpi∂Φ∂pi−p¯3μp3∂Φ∂p3.It is a straightforward but lengthy computation to show that the special (non anomalous) conformal transformation in d dimension takes the form, for the scalar component(4.4)KscalarκΦ=0 with(4.5)Kscalarκ=∑i=13piκKi(4.6)Ki≡∂2∂pi∂pi+d+1−2Δipi∂∂pi with the expression (4.5) which can be split into the two independent equations(4.7)∂2Φ∂pi∂pi+1pi∂Φ∂pi(d+1−2Δ1)−∂2Φ∂p3∂p3−1p3∂Φ∂p3(d+1−2Δ3)=0i=1,2. Notice that in the derivation of (4.5) one needs at an intermediate step the derivative of the scaling WI(4.8)p1∂2Φ∂p3∂p1+p2∂2Φ∂p3∂p2=(Δ−2d−1)∂Φ∂p3−p3∂2Φ∂p3∂p3. Defining(4.9)Kij≡Ki−Kj Eqs. (4.7) take the form(4.10)K13κΦ=0andK23κΦ=0.5Transverse Ward identitiesTo fix the form of the correlator we need to impose the transverse WI on the vector lines and the conservation WI for Tμν. In this section we briefly discuss their derivation and their explicit expressions. We consider the functional(5.1)W[g,A]=∫Dψ¯Dψe−(S0[g,ψ]+S1[A,ψ]) integrated over the fermions ψ, in the background of the metric gμν and of the gauge field Aμa. In the case of a nonabelian gauge theory the action is given by(5.2)S0[g,A,ψ]=−14∫d4x−gxFμνaFμνa+∫d4x−gxiψ¯γμDμψS1[g,A]=∫−gxJμaAμa with(5.3)Fμνa=∇μAνa−∇νAμa+gcfabcAμbAνc=∂μAνa−∂νAμa+gcfabcAμbAνc,∇μAνa=∂μAνa+ΓμνλAλa with Jμa=gcψ¯γμTaψ denoting the fermionic current, with Ta the generators of the theory and ∇μ denoting the covariant derivative in the curved background on a vector field. The local Lorentz and gauge covariant derivative (D) on the fermions acts via the spin connection(5.4)Dμψ=(∂μψ+AμaTa+14ωμa_b_γa_b_)ψ having denoted with a_b_ the local Lorentz indices. A local Lorentz covariant derivative (D) can be similarly defined for a vector field, say Va_, via the Vielbein eμa_ and its inverse ea_μ(5.5)DμVa_=∂μVa_+ωμb_a_Vb_ with(5.6)∇μVρ=ea_ρDμVa_ with the Christoffel and the spin connection related via the holonomic relation(5.7)Γμνρ=ea_ρ(∂μeνa_+ωa_μb_eνb_). Diffeomorphism invariance of the generating functional (5.1) gives(5.8)∫ddx(δWδgμνδgμν(x)+δWδAμaδAμa(x))=0 where the variation of the metric and the gauge fields are the corresponding Lie derivatives, for a change of variables xμ→xμ+ϵμ(x)(5.9)δAμa(x)=−∇αAμaϵα−Aαa∇μϵαδgμν=−∇μϵν−∇νϵμ while for a gauge transformation with a parameter θa(x)(5.10)δAμa=D_μθa≡∂μθa+gcfabcAμbθc. Using (5.9), Eq. (5.8) becomes(5.11)0=〈∫d4x(δ(S0+S1)δgμνδgμν+δS1δAμaδAμa)〉=〈∫d4x−gx[∇μTμν+(∇μAνa−∇νAμa)Jμa+∇μJμaAνa]ϵν(x)〉 while the condition of gauge invariance gives(5.12)∫ddxδWδAμaδAμa=〈∫d4x−gxJaμD_μθa〉=0 which, in turn, after an integration by parts, generates the gauge WI(5.13)〈∇μJμa〉=gcfabc〈Jμb〉Aμc. Inserting this relation into (5.11) we obtain the conservation WI(5.14)〈∇μTμν〉+Fμνa〈Jμa〉=0. In the abelian case, diffeomorphism and gauge invariance then give the relations(5.15)0=∇ν〈Tμν〉+Fμν〈Jν〉0=∇ν〈Jν〉 with naive scale invariance gives the traceless condition(5.16)gμν〈Tμν〉=0. The functional differentiation of (5.15) and (5.16) allows to derive ordinary Ward identities for the various correlators. In the TJJ case we obtain, after a Fourier transformation, the conservation equation(5.17)p1ν1〈Tμ1ν1(p1)Jμ2(p2)Jμ3(p3)〉=4[δμ1μ2p2λ〈Jλ(p1+p2)Jμ3(p3)〉−p2μ1〈Jμ2(p1+p2)Jμ3(p3)〉]+4[δμ1μ3p3λ〈Jλ(p1+p3)Jμ2(p2)〉−p3μ1〈Jμ3(p1+p3)Jμ2(p2)〉] and vector current Ward identities(5.18)p2μ2〈Tμ1ν1(p1)Jμ2(p2)Jμ3(p3)〉=0(5.19)p3μ3〈Tμ1ν1(p1)Jμ2(p2)Jμ3(p3)〉=0, while the naive identity (5.16) gives the non-anomalous condition(5.20)δμ1ν1〈Tμ1ν1(p1)Jμ2(p2)Jμ3(p3)〉=0, valid in the d≠4 case. We recall that the 2-point function of two conserved vector currents Ji (i=2,3) [24] in any conformal field theory in d dimension is given by(5.21)〈J2α(p)J3β(−p)〉=δΔ2Δ3(c123ΓJ)παβ(p)(p2)Δ2−d/2,ΓJ=πd/24Δ2−d/2Γ(d/2−Δ2)Γ(Δ2), with c123 an overall constant and Δ2=d−1. In our case Δ2=Δ3=d−1 and Eq. (5.17) then takes the form(5.22)p1μ1〈Tμ1ν1(p1)Jμ2(p2)Jμ3(p3)〉=4c123ΓJ(δν1μ2p2λ(p32)d/2−Δ2πλμ3(p3)−p2ν1(p32)d/2−Δ2πμ2μ3(p3)+δν1μ3p3λ(p22)d/2−Δ2πλμ2(p2)−p3ν1(p22)d/2−Δ2πμ3μ2(p2)). Explicit expressions of the secondary CWI's are determined using (5.18) and (5.20) and the explicit form (5.22).6TJJ reconstruction the BMS wayWe are now going investigate the BMS approach, which is technically quite involved, highlighting several steps which are crucial in order to clarify the basic structure of the method. The method is exemplified in the case of the TJJ. Several intermediate steps, which we believe are necessary in order to characterize the approach, have been worked out independently and are based on the use of the Lorentz Ward identities.Given the partial symmetry of the TJJ correlator, for instance, respect to the TTT case, one can choose as independent momenta either p1 and p2 or, more conveniently, p2 and p3, given the symmetry of the two J currents.With the first choice, outlined below, the current J(p3) is singlet under the (spin) Lorentz generators. With the second choice, the two currents are treated symmetrically and the stress energy tensor is treated as a singlet under the same generators. The derivation of the CWI's in this second case will be outlined in section 7. The equations obtained in the two cases are obviously the same.First of all we discuss the canonical Ward identities for the 〈TJJ〉 correlation function in momentum space. From the general definition of the global Ward identities in position space(6.1)∑j=13Gg(xj)〈Tμ1ν1(x1)Jμ2(x2)Jμ3(x3)〉=0, where Gg is the generator of the infinitesimal symmetry transformation. The dilatation Ward identities take the form(6.2)0=[∑j=13Δj−(n−1)d−∑j=12pjα∂∂pjα]〈Tμ1ν1(p1)Jμ2(p2)Jμ3(p¯3)〉. To proceed towards the analysis of the constraints, it is essential to introduce the Lorentz covariant Ward identities(6.3)0=∑j=12[pjν∂∂pjμ−pjμ∂∂pjν]〈Tμ1ν1(p1)Jμ2(p2)Jμ3(p¯3)〉+2(δα1νδμ(μ1−δα1μδν(μ1)〈Tν1)α1(p1)Jμ2(p2)Jμ3(p¯3)〉+(δα2νδμμ2−δμα2δνμ2)〈Tμ1ν1(p1)Jα2(p2)Jμ3(p¯3)〉+(δα3νδμμ3−δμα3δνμ3)〈Tμ1ν1(p1)Jμ2(p2)Jα3(p¯3)〉, where(6.4)δν(μ1〈Tν1)α1Jμ2Jμ3〉≡12(δνμ1〈Tν1α1Jμ2Jμ3〉+δνν1〈Tμ1α1Jμ2Jμ3〉), and finally the special conformal Ward identities(6.5)0=∑j=12[2(Δj−d)∂∂pjκ−2pjα∂∂pjα∂∂pjκ+(pj)κ∂∂pjα∂∂pjα]×〈Tμ1ν1(p1)Jμ2(p2)Jμ3(p¯3)〉+4(δκ(μ1∂∂p1α1−δα1κδλ(μ1∂∂p1λ)〈Tν1)α1(p1)Jμ2(p2)Jμ3(p¯3)〉+2(δκμ2∂∂p2α2−δα2κδλμ2∂∂p2λ)〈Tμ1ν1(p1)Jα2(p2)Jμ3(p¯3)〉, which we will use in the next sections in order to determine the tensor structure of this correlator.6.1ProjectorsThe basic observation in the BMS approach is that the action of the special conformal, trace and conservation (longitudinal) WI's take a simpler form if we enforce a decomposition of the tensor correlators in terms of transverse traceless, longitudinal and trace parts and project the Ward identities on the same subspaces. Recall that for a symmetric tensor such as the EMT, this decomposition is performed using the following projectors(6.6)παμ=δαμ−pμpαp2,π˜αμ=1d−1παμ(6.7)Παβμν=12(παμπβν+πβμπαν)−1d−1πμνπαβ,(6.8)Iαβμν=1p2pβ(pμδαν+pνδαμ−pαpβp2(δμν+(d−2)pμpνp2))(6.9)Lαβμν=12(Iαβμν+Iβαμν)ταβμν=π˜μνδαβ with(6.10)δαβμν=Παβμν+Σαβμν(6.11)Σαβμν≡Lαβμν+ταβμν The previous identities allow to decompose a symmetric tensor into its transverse traceless (via Π), longitudinal (via L) and trace parts (via τ), or on the sum of the combined longitudinal and trace contributions (via Σ). We are now going to illustrate the approach in the case of the TTT correlation function. The transverse traceless projections will be denoted as (t) and the trace parts with (s). We will be denoting such correlator as ϕ and try to resort to a condensed notation in order to characterize the algebraic structure of the procedure. For a rank-6 correlator of 3 T's, ϕ will denote the tensor(6.12)ϕ≡〈Tμ1ν1(p1)Tμ2ν2(p2)Tμ3ν3(p3)〉. We can act on this correlator with the two projectors Π and Σ on each (combined) pair of indices and momenta. For instance, the transverse traceless part of ϕ is obtained by acting with 3 Π projectors on the indices of the EMT's(6.13)ϕttt≡Π1Π2Π3ϕ(6.14)≡Π1α1β1μ1ν1Π2α2β2μ2ν2Π3α3β3μ3ν3〈Tα1β1(p1)Tα2β2(p2)Tα3β3(p3)〉, where Π1α1β1μ1ν1≡Πα1β1μ1ν1(p1). Similarly, the remaining 7 components of the ϕ correlator can be obtained by acting with all the other combinations of projectors (Σ,Π), to obtain(6.15)ϕ=ϕttt+ϕtst+ϕtss+ϕsss+ϕstt+ϕsst+ϕsts. Acting with the special conformal transformation Kκ (both with the scalar and the spin parts) on ϕ we can again project the result onto the orthogonal subspaces ttt,tss, etc., and try to solve the equations separately in each of these 8 sectors. In our condensed notation the equation for the special conformal transformation takes the form(6.16)ϕ′=Kκϕ=0, and its projection into the 8 independent sectors, such as, for instance(6.17)ϕttt′≡Π1Π2Π3Kκϕ=0,(6.18)ϕstt′≡Σ1Π2Π3Kκϕ=0,ϕsst′≡Σ1Σ2Π3Kκϕ=0,… and so on, can be obtained by the action of the Π's and Σ's on (6.16). It is important to realize that only the equation ϕttt′=0 involves 3-point functions beside 2 point functions, and needs to be solved. The remaining sectors do not give any new equation, since they involve only 2-point functions, being related to the conservation and trace WI's. However they define consistency conditions for lower point functions that will introduce some constraint on the arbitrary constants appearing in the solutions of the primary WI's.To illustrate these points we will treat the correlators ϕ,ϕ′ as vectors in a functional space on which the Kκ operator will act both in differential form and algebraically via its spin rotation matrices. For instance we define(6.19)ϕttt≡Ptttϕ,wherePttt≡Π1Π2Π3,ϕtst≡PtstϕwithPtst≡Π1Σ2Π3,… and similarly in the other cases. In general, it is convenient to characterize the action of Kκ on each subspace via a projection, such as(6.20)ϕttt′(ttt)≡PtttKκPtttϕϕstt′(ttt)=PsttKκPtttϕ,… and so on, for a total of 64=8×8 sectors. In the first expressions above, for example, the original transverse traceless projection (ttt) is acted upon by Kκ and then it is re-projected onto the ttt sector. There are several simplifications among these matrix elements. For example, a direct computation, that we will prove below, gives(6.21)PsttKκPtttϕ=0,PtstKκPtttϕ=0,PttsKκPtttϕ=0,… showing that the ϕttt amplitude is mapped only into another amplitude in the same ttt subspace by the action of Kκ.6.1.1Endomorphic action of Kκ on the transverse-traceless sectorKκ acts as endomorphism on the transverse traceless sector of a tensor correlator. To illustrate this point we consider the case of the TTT, though the approach is generic. Define(6.22)Yμ1ν1μ2ν2μ3ν3=Π1α1β1μ1ν1Π2α2β2μ2ν2Π3α3β3μ3ν3〈Tα1β1Tα2β2Tα3β3〉 to be the transverse traceless projection of the TTT. One can check the transversality of the action of K1scalarκ (the other contribution of Kκ being similar) by contracting Y with p1(6.23)p1μ1K1scalarκYμ1ν1μ2ν2μ3ν3=(−2p1αp1μ1∂2∂p1α∂p1κ+p1κp1μ1∂2∂p1α∂p1α)Yμ1ν1μ2ν2μ3ν3=−2p1α∂∂p1α(p1μ1∂∂p1κYμ1ν1μ2ν2μ3ν3)+2p1μ1∂p1κYμ1ν1μ2ν2μ3ν3+p1κ∂∂p1α(p1μ1∂p1αYμ1ν1μ2ν2μ3ν3)−p1κ∂∂p1μ1Yμ1ν1μ2ν2μ3ν3=2p1α∂∂p1κYκν1μ2ν2μ3ν3−2Yκν1μ2ν2μ3ν3−2p1κ∂∂p1μ1Yμ1ν1μ2ν2μ3ν3 where we have rearranged the partial derivatives. For the spin part we obtain(6.24)p1μ1K1spinκYμ1ν1μ2ν2μ3ν3=2p1μ1(δκμ1∂∂p1α−δκα∂p1μ1)Yαν1μ2ν2μ3ν3+2p1μ1(δκν1∂∂p1α−δκα∂p1ν1)Yμ1αμ2ν2μ3ν3=2p1κ∂∂p1αYαν1μ2ν2μ3ν3−2p1μ1∂∂p1μ1Yκν1μ2ν2μ3ν3+2Yκν1μ2ν2μ3ν3. Adding (6.23) and (6.24) it is shown that(6.25)p1μ1K1κYμ1ν1μ2ν2μ3ν3=0, which clearly holds for the entire K operator since Π1 filters to the left of K2, obtaining(6.26)p1μ1KκYμ1ν1μ2ν2μ3ν3=0. Notice that in the derivation of this result the nonlinear character of the action of Kscalarκ, which induces mixed derivative terms does not play any role. Due to the trace-free property of the projectors, then we obtain in our condensed notation(6.27)Σ1Kκϕttt=0. The solution of the CWI's are then constructed, in this method, by acting on the entire correlator having parametrized its transverse traceless parts in therms of a minimal set of form factors plus trace/longitudinal terms (the semilocal or pinched terms). Semilocal terms are those containing a single delta function which will pinch two of the three external coordinates. The term ultralocal (or local) refers to the contribution of the anomaly itself, which is obtained when all the 3 point of the correlator coalesce.6.2Application to the TJJTurning to the TJJ case, we can divide the 3-point function into two parts: the transverse-traceless part and the semi-local part (indicated by subscript loc) expressible through the transverse and trace Ward Identities. These parts are obtained by using the projectors Π and Σ, previously defined. We can then decompose the full 3-point function as follows(6.28)〈Tμ1ν1Jμ2Jμ3〉=〈tμ1ν1jμ2jμ3〉+〈Tμ1ν1Jμ2jlocμ3〉+〈Tμ1ν1jlocμ2Jμ3〉+〈tlocμ1ν1Jμ2Jμ3〉−〈Tμ1ν1jlocμ2jlocμ3〉−〈tlocμ1ν1jlocμ2Jμ3〉−〈tlocμ1ν1Jμ2jlocμ3〉+〈tlocμ1ν1jlocμ2jlocμ3〉. All the terms on the right-hand side, apart from the first one, may be computed by means of transverse and trace Ward Identities. The exact form of the Ward identities depends on the exact definition of the operators involved, but more importantly, all these terms depend on 2-point function only. The main goal now is to write the general form of the transverse-traceless part of the correlator and to give the solution using the Conformal Ward identities.Using the projectors Π and π one can write the most general form of the transverse-traceless part as(6.29)〈tμ1ν1(p1)jμ2(p2)jμ3(p3)〉=Πα1β1μ1ν1(p1)πα2μ2(p2)πα3μ3(p3)Xα1β1α3α3, where Xα1β1α3α3 is a general tensor of rank four built from the metric and momenta. We can enumerate all possible tensor that can appear in Xα1β1α3α3 preserving the symmetry of the correlator, as illustrated in [14](6.30)〈tμ1ν1(p1)jμ2(p2)jμ3(p3)〉=Π1α1β1μ1ν1π2α2μ2π3α3μ3(A1p2α1p2β1p3α2p1α3+A2δα2α3p2α1p2β1+A3δα1α2p2β1p1α3+A3(p2↔p3)δα1α3p2β1p3α2+A4δα1α3δα2β1), where we have used the symmetry properties of the projectors, and the coefficients Aii=1,…,4 are the form factors, functions of p12,p22 and p32. This ansatz introduces a minimal set of form factors which will be later determined by the solutions of the CWI's. For future discussion, we will refer to this basis as to the A-basis.We can now consider the dilatation Ward identities for the transverse-traceless part obtained by the decomposition of (6.2). We are then free to apply the projectors Π and π to this decomposition in order to obtain the final result(6.31)0=Πα1β1μ1ν1(p1)πα2μ2(p2)πα3μ3(p3)[∑j=13Δj−2d−∑j=12pjα∂∂pjα][A1p2α1p2β1p3α2p1α3+A2δα2α3p2α1p2β1+A3δα1α2p2β1p1α3+A3(p2↔p3)δα1α3p2β1p3α2+A4δα1α3δα2β1]. It is possible to obtain from this projection a set of differential equations for all the form factors. These equations are expressed as(6.32)[2d+Nn−∑j=13Δj+∑j=12pjα∂∂pjα]An(p1,p2,p3)=0, where Nn is the tensorial dimension of An, i.e. the number of momenta multiplying the form factor An and the projectors Π and π.Turning to the special CWI's, 〈TJJ〉 in (6.5), we can write the same equation in the form(6.33)Kκ〈Tμ1ν1(p1)Jμ2(p2)Jμ3(p3)〉=0, where Kκ is the special conformal generator. As before, we introduce the decomposition of the 3-point function to obtain(6.34)0=Kκ[〈tμ1ν1jμ2jμ3〉+〈tlocμ1ν1jμ2jμ3〉+〈tμ1ν1jlocμ2jμ3〉+〈tμ1ν1jμ2jlocμ3〉+〈tlocμ1ν1jlocμ2jμ3〉+〈tlocμ1ν1jlocμ2jμ3〉+〈tμ1ν1jlocμ2jlocμ3〉+〈tlocμ1ν1jlocμ2jlocμ3〉]. In order to isolate the equations for the form factors appearing in the decomposition, we are free to apply the projectors Π and π defined previously. Through a lengthy calculation we find(6.35)Πμ1ν1ρ1σ1(p1)πμ2ρ2(p2)πμ3ρ3(p3)Kκ〈tlocμ1ν1jμ2jμ3〉=Πμ1ν1ρ1σ1πμ2ρ2πμ3ρ3[4dp12δκμ1p1α1〈Tα1ν1Jμ2Jμ3〉]Πμ1ν1ρ1σ1(p1)πμ2ρ2(p2)πμ3ρ3(p3)Kκ〈tμ1ν1jlocμ2jμ3〉=Πμ1ν1ρ1σ1πμ2ρ2πμ3ρ3[2(d−2)p22δκμ2p2α2〈Tα1ν1Jα2Jμ3〉]Πμ1ν1ρ1σ1(p1)πμ2ρ2(p2)πμ3ρ3(p3)Kκ〈tμ1ν1jμ2jlocμ3〉=Πμ1ν1ρ1σ1πμ2ρ2πμ3ρ3[2(d−2)p32δκμ3p3α3〈Tα1ν1Jμ2Jα3〉] and all the terms with at least two insertion of local terms are zero. T We have verified, as expected, that the equations above remain invariant if we choose as independent momenta p2 and p3 while acting on p1 indirectly by the derivative chain rule. More details on this analysis will be given in a section below. In this way we may rewrite (6.5) in the form(6.36)0=Πμ1ν1ρ1σ1(p1)πμ2ρ2(p2)πμ3ρ3(p3)(Kκ〈Tμ1ν1(p1)Jμ2(p2)Jμ3(p3)〉)=Πμ1ν1ρ1σ1(p1)πμ2ρ2(p2)πμ3ρ3(p3){Kκ〈tμ1ν1(p1)jμ2(p2)jμ3(p3)〉+4dp12δκμ1p1α1〈Tα1ν1(p1)Jμ2(p2)Jμ3(p3)〉+2(d−2)p22δκμ2p2α2〈Tα1ν1Jα2Jμ3〉+2(d−2)p32δκμ3p3α3〈Tα1ν1Jμ2Jα3〉}. The equation above is an independent derivation of the corresponding BMS result, which is not offered in [14]. Notice that our derivation, which details the various contributions coming from the local terms in the TJJ, has been derived using heavily the Lorentz Ward identities.The last three terms may be re-expressed in terms of 2-point functions via the transverse Ward identities. After other rather lengthy computations, we find that the first term in the previous expression, corresponding to the transverse traceless contributions, can be written in the form(6.37)Πμ1ν1ρ1σ1(p1)πμ2ρ2(p2)πμ3ρ3(p3)[Kκ〈tμ1ν1(p1)jμ2(p2)jμ3(p3)〉]=Πμ1ν1ρ1σ1(p1)πμ2ρ2(p2)πμ3ρ3(p3)××[p1κ(C11p1μ3p2μ1p2ν1p3μ2+C12δμ2μ3p2μ1p2ν1+C13δμ1μ2p2ν1p1μ3+C14δμ1μ3p2ν1p3μ2+C15δμ1μ2δν1μ3)+p2κ(C21p1μ3p2μ1p2ν1p3μ2+C22δμ2μ3p2μ1p2ν1+C23δμ1μ2p2ν1p1μ3+C24δμ1μ3p2ν1p3μ2+C25δμ1μ2δν1μ3)+δμ1κ(C31p1μ3p2ν1p3μ2+C32δμ2μ3p2ν1+C33δμ2ν1p1μ3+C34δμ3ν1p3μ2)+δμ2κ(C41p1μ3p2μ1p2ν1+C42δμ1μ3p2ν1)+δμ3κ(C51p3μ2p2ν1p3μ2+C52δμ1μ2p2ν1)] where now Cij are differential equations involving the form factors A1,A2,A3,A4 of the representation of the 〈tjj〉 in (6.30). For any 3-point function, the resulting equations can be divided into two groups, the primary and the secondary conformal Ward identities. The primary are second-order differential equations and appear as the coefficients of transverse or transverse-traceless tensor containing p1κ and p2κ, where κ is the special index related to the conformal operator Kκ. The remaining equations, following from all other transverse or transverse-traceless terms, are then secondary conformal Ward identities and are first-order differential equations.6.3Primary CWI'sFrom (6.36) and (6.37) one finds that the primary CWI's are equivalent to the vanishing of the coefficients C1j and C2j for j=1,…,5. The CWI's can be rewritten in terms of the operators defined in Eq. (4.9) as(6.38)0=C11=K13A10=C12=K13A2+2A10=C13=K13A3−4A10=C14=K13A3(p2↔p3)0=C15=K13A4−2A3(p2↔p3)0=C21=K23A10=C22=K23A20=C23=K23A3−4A10=C24=K23A3(p2↔p3)+4A10=C25=K23A4+2A3−2A3(p2↔p3)6.4Secondary CWI'sThe secondary conformal Ward identities are first-order partial differential equations and in principle involve the semi-local information contained in jlocμ and tlocμν. In order to write them compactly, one defines the two differential operators(6.39)LN=p1(p12+p22−p32)∂∂p1+2p12p2∂∂p2+[(2d−Δ1−2Δ2+N)p12+(2Δ1−d)(p32−p22)](6.40)R=p1∂∂p1−(2Δ1−d). The reason for introducing such operators comes from (6.37), once the action of Kκ is made explicit. The separation between the two sets of constraints comes from the same equation, and in particular from the terms trilinear in the momenta within the square bracket. One needs also the symmetric versions of such operators(6.41)LN′=LN,withp1↔p2andΔ1↔Δ2,(6.42)R′=R,withp1↦p2andΔ1↦Δ2. 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 〈TJJ〉 case one finds considering the structure of Eqs. (6.36) and (6.37)(6.43)C31=−2p12[L4A1+RA3−RA3(p2↔p3)]C32=−2p12[L2A2−p12(A3−A3(p2↔p3))]C33=−1p12[L4A3−2RA4]C34=−1p12[L4A3(p2↔p3)+2RA4−4p12A3(p2↔p3)](6.44)C41=1p22[L3′A1−2R′A2+2R′A3]C42=1p22[L1′A3(p2↔p3)+p22(4A2−2A3)+2R′A4]C51=1p3[(L4−L3′)A1−2(2d+R+R′)A2+2(2d+R+R′)A3(p2↔p3)]C52=1p32[(L2−L1′)A3−4p32A2+2p32A3(p2↔p3)+2(2d−2+R+R′)A4] From (6.36) and (6.37) using (5.22) the secondary CWI's take the explicit form(6.45)C31=C41=C42=C51=C52=0,C32=16dc123ΓJp12[1(p32)σ0−1(p22)σ0],C33=16dc123ΓJp12(p32)σ0,C34=−16dc123ΓJp12(p22)σ0, where in our σ0=d/2−Δ2. Expressed in this form all the scalar equations for the Ai are not apparently symmetric in the exchange of p2 and p3, and it may not be immediately evident that they can be recast in such a way that the symmetry is respected.7Symmetric treatment of the J currentsLet's now consider p1 as dependent momentum, showing the equivalence of the CWI's with this second choice. As we have just mentioned above, this choice is the preferred one in the search for the solutions of the TJJ. In this case, the action of the spin (Lorentz) part of the transformation will leave the stress energy tensor as a singlet, acting implicitly on p1 via the chain rule. As we are going to show, the resulting equations will be linear combinations of the original part. This extends the analysis presented by BMS.The structure of the decomposition in (6.30) of the 〈TJJ〉 correlator is still valid but now the explicit form of the special conformal operator Kκ has to be modified as(7.1)Kκ〈Tμ1ν1Jμ2Jμ3〉=∑j=23[2(Δj−d)∂∂pjκ−2pjα∂∂pjα∂∂pjκ+(pj)κ∂∂pjα∂∂pjα]×〈Tμ1ν1(p¯1)Jμ2(p2)Jμ3(p3)〉+2(δκμ2∂∂p2α2−δα2κδλμ2∂∂p2λ)〈Tμ1ν1(p¯1)Jα2(p2)Jμ3(p3)〉+2(δκμ3∂∂p3α3−δα3κδλμ3∂∂p3λ)〈Tμ1ν1(p¯1)Jμ2(p2)Jα3(p3)〉 where p¯1μ=−p2μ−p3μ. Considering the SCWI's for the 3-point function we can writeKκ(p2,p3)〈Tμ1ν1(p¯1)Jμ2(p2)Jμ3(p3)〉=0, in which we have stress the p2 and p3 dependence of the special conformal operator. Then one has to take the decomposition of the 3-point function as in (6.30) and using the relations (6.35), that are still valid in this case, one derives (6.36), in which now the K operator is defined in terms of p2 and p3 only. As in the previous case, one finds CWI's which are similar to those given in (6.37)(7.2)Πμ1ν1ρ1σ1(p1)πμ2ρ2(p2)πμ3ρ3(p3)[Kκ〈tμ1ν1(p1)jμ2(p2)jμ3(p3)〉]=Πμ1ν1ρ1σ1(p1)πμ2ρ2(p2)πμ3ρ3(p3)××[p2κ(C˜11p1μ3p2μ1p2ν1p3μ2+C˜12δμ2μ3p2μ1p2ν1+C˜13δμ1μ2p2ν1p1μ3+C˜14δμ1μ3p2ν1p3μ2+C˜15δμ1μ2δν1μ3)+p3κ(C˜21p1μ3p2μ1p2ν1p3μ2+C˜22δμ2μ3p2μ1p2ν1+C˜23δμ1μ2p2ν1p1μ3+C˜24δμ1μ3p2ν1p3μ2+C˜25δμ1μ2δν1μ3)+δμ1κ(C˜31p1μ3p2ν1p3μ2+C˜32δμ2μ3p2ν1+C˜33δμ2ν1p1μ3+C˜34δμ3ν1p3μ2)+δμ2κ(C˜41p1μ3p2μ1p2ν1+C˜42δμ1μ3p2ν1)+δμ3κ(C˜51p3μ2p2ν1p3μ2+C˜52δμ1μ2p2ν1)]. In this case we obtain the primary WI's by imposing the vanishing of the coefficients C˜ij, for i=1,2 and j=1,…,5. In this way we get(7.3)0=C˜11=K21A10=C˜12=K21A2−2A10=C˜13=K21A30=C˜14=K21A3(p2↔p3)+4A10=C˜15=K21A4+2A30=C˜21=K31A10=C˜22=K31A2−2A10=C˜23=K31A3+4A10=C˜24=K31A3(p2↔p3)0=C˜25=K31A4+2A3(p2↔p3) and it is simple to verify that these equations are equivalent to those given in (6.38). In the case of the secondary WI's we have to consider some further properties of the form factors. For instance the coefficient C˜31 has the explicit form(7.4)C˜31=2p12[p2(p12−p22+p32)∂∂p2A1−p12p3∂∂p3A1−p22p3∂∂p3A1+p33∂∂p3A1−p2∂∂p2A3−p3∂∂p3A3+p2∂∂p2A3(p2↔p3)+p3∂∂p3A3(p2↔p3)−6(p22−p32)A1−4(A3−A3(p2↔p3))] in which it is possible to substitute the derivative with respect to p3 in terms of derivatives with respect to p2 and p1 using the dilatation Ward identities(7.5)∂∂p3An=1p3[(d−2−Nn)An−∑j=12pj∂∂pjAn]. Using the identity given above in (7.4), one derives the relation(7.6)C˜31=2p12[L4A1+RA3−RA3(p2↔p3)] with the identification of the differential operators L and R defined in (6.39) and (6.40). In this way it is possible to show that all the coefficients related to the secondary Ward identities are the same of those obtained with p3 as the dependent momentum. This argument proves that in spite of the choice of the dependent momentum, the scalar equations for the form factors related to the CWI's remain identical.8The Fuchsian approach to the solutions of the primary CWI's and universalityIn this section we are going to investigate the Fuchsian structure of the equations. The goal of the section is to present a new method of solution which differs from the one based on 3K integrals presented in [14]. We should mention that the number of integration constants introduced by the primary CWI's, using this method, may not necessarily coincide with those presented in [14], and the constraints imposed by the secondary CWIs, that we will not discuss, will obviously be different.The goal of this section is twofold. We want to show first of all that the Fuchsian exponents (defined as (ai,bj) below), are universal and characterize the entire system of equations. In the scalar case, as well as for all the 3-point functions that we have investigated, we have verified that always the same set of exponents (ai,bj) are generated.The second important feature is that the method allows to characterize particular solutions of 4 and higher point functions in some restricted kinematics, allowing a significant generalization of the analysis presented here, with new special functions appearing in the solutions. Details of this study will be presented in a separate work.Being the CWI's a system of equations, we will first solve for each of the form factors, starting from the equations for A1, which are homogeneous, and then proceed towards the inhomogeneous ones, from A2 to A4. For each form factor we identify the general solution and a particular solution, which are added together. Then we impose the symmetry constraints on the two vector lines, due to Bose symmetry. For example, the solution for A2 will constraint the constants appearing in the general solution of A1, and so on for A3 and A4. The independent constants of integration are identified only at the end, once all the constraints from A1 to A4 are put together. We have included a small section where we summarize the final expressions of the form factors by this method.8.1Scalar 3-point functionsTo illustrate our approach we start reviewing the case of the scalar correlator Φ(p1,p2,p3), which is simpler, defined by the two homogeneous conformal equations(8.1)K31Φ=0K21Φ=0 combined with the scaling equation(8.2)∑i=13pi∂∂piΦ=(Δ−2d)Φ. Following the approach presented in [24], the ansatz for the solution can be taken of the form(8.3)Φ(p1,p2,p3)=p1Δ−2dxaybF(x,y) with x=p22p12 and y=p32p12. Here we are taking p1 as “pivot” in the expansion, but we could equivalently choose any of the 3 momentum invariants. Φ is required to be homogenous of degree Δ−2d under a scale transformation, according to (8.2), and in (8.3) this is taken into account by the factor p1Δ−2d. The use of the scale invariant variables x and y takes to the hypergeometric form of the solution. One obtains(8.4)K21ϕ=4p1Δ−2d−2xayb(x(1−x)∂∂x∂x+(Ax+γ)∂∂x−2xy∂2∂x∂y−y2∂2∂y∂y+Dy∂∂y+(E+Gx))F(x,y)=0 with(8.5)A=D=Δ2+Δ3−1−2a−2b−3d2γ(a)=2a+d2−Δ2+1G=a2(d+2a−2Δ2)E=−14(2a+2b+2d−Δ1−Δ2−Δ3)(2a+2b+d−Δ3−Δ2+Δ1). Similar constraints are obtained from the equation K31Φ=0, with the obvious exchanges (a,b,x,y)→(b,a,y,x)(8.6)K31ϕ=4p1Δ−2d−2xayb(y(1−y)∂∂y∂y+(A′y+γ′)∂∂y−2xy∂2∂x∂y−x2∂2∂x∂x+D′x∂∂x+(E′+G′y))F(x,y)=0 with(8.7)A′=D′=Aγ′(b)=2b+d2−Δ3+1G′=b2(d+2b−2Δ3)E′=E. Notice that in (8.6) we need to set G/x=0 in order to perform the reduction to the hypergeometric form of the equations, which implies that(8.8)a=0≡a0ora=Δ2−d2≡a1. From the equation K31Φ=0 we obtain a similar condition for b by setting G′/y=0, thereby fixing the two remaining indices(8.9)b=0≡b0orb=Δ3−d2≡b1. The four independent solutions of the CWI's will all be characterized by the same 4 pairs of indices (ai,bj) (i,j=1,2). Setting(8.10)α(a,b)=a+b+d2−12(Δ2+Δ3−Δ1)β(a,b)=a+b+d−12(Δ1+Δ2+Δ3) then(8.11)E=E′=−α(a,b)β(a,b)A=D=A′=D′=−(α(a,b)+β(a,b)+1), the solutions take the form(8.12)F4(α(a,b),β(a,b);γ(a),γ′(b);x,y)=∑i=0∞∑j=0∞(α(a,b),i+j)(β(a,b),i+j)(γ(a),i)(γ′(b),j)xii!yjj! where (α,i)=Γ(α+i)/Γ(α) is the Pochammer symbol. We will refer to α…γ′ as to the first, …, fourth parameters of F4.The 4 independent solutions are then all of the form xaybF4, where the hypergeometric functions will take some specific values for its parameters, with a and b fixed by (8.8) and (8.9). Specifically we have(8.13)Φ(p1,p2,p3)=p1Δ−2d∑a,bc(a,b,Δ→)xaybF4(α(a,b),β(a,b);γ(a),γ′(b);x,y) where the sum runs over the four values ai,bi i=0,1 with arbitrary constants c(a,b,Δ→), with Δ→=(Δ1,Δ2,Δ3). Notice that (8.13) is a very compact way to write down the solution. However, once this type of solutions of a homogeneous hypergeometric system are inserted into an inhomogeneous system of equations, the sum over a and b needs to be made explicit. For this reason it is convenient to define(8.14)α0≡α(a0,b0)=d2−Δ2+Δ3−Δ12,β0≡β(b0)=d−Δ1+Δ2+Δ32,γ0≡γ(a0)=d2+1−Δ2,γ0′≡γ(b0)=d2+1−Δ3, to be the 4 basic (fixed) hypergeometric parameters, and define all the remaining ones by shifts respect to these. The 4 independent solutions can be re-expressed in terms of the parameters above as(8.15)S1(α0,β0;γ0,γ0′;x,y)≡F4(α0,β0;γ0,γ0′;x,y)=∑i=0∞∑j=0∞(α0,i+j)(β0,i+j)(γ0,i)(γ0′,j)xii!yjj! and(8.16)S2(α0,β0;γ0,γ0′;x,y)=x1−γ0F4(α0−γ0+1,β0−γ0+1;2−γ0,γ0′;x,y),(8.17)S3(α0,β0;γ0,γ0′;x,y)=y1−γ0′F4(α0−γ0′+1,β0−γ0′+1;γ0,2−γ0′;x,y),(8.18)S4(α0,β0;γ0,γ0′;x,y)=x1−γ0y1−γ0′F4(α0−γ0−γ0′+2,β0−γ0−γ0′+2;2−γ0,2−γ0′;x,y). Notice that in the scalar case, one is allowed to impose the complete symmetry of the correlator under the exchange of the 3 external momenta and scaling dimensions, as discussed in [24]. This reduces the four constants to just one. We are going first to extend this analysis to the case of the A1−A4 form factors of the TJJ.8.2Form factors: the solution for A1The solutions for the form factors A1−A4 can be derived using a similar, but modified approach, being the equations also inhomogeneous. As previously we take as a pivot p12, and assume a symmetry under the (P23) exchange of (p2,Δ2) with (p3,Δ3) in the correlator. In the case of two photons Δ2=Δ3=d−1.We start from A1 by solving the two equations from (6.38)(8.19)K21A1=0K31A1=0. In this case we introduce the ansatz(8.20)A1=p1Δ−2d−4xaybF(x,y) and derive two hypergeometric equations, which are characterized by the same indices (ai,bj) as before in (8.8) and (8.9), but new values of the 4 defining parameters. We obtain(8.21)A1(p1,p2,p3)=p1Δ−2d−4∑a,bc(1)(a,b,Δ→)xaybF4(α(a,b)+2,β(a,b)+2;γ(a),γ′(b);x,y) with the expression of α(a,b),β(a,b),γ(a),γ′(b) as given before, with the obvious switching of the Δi in order to comply with the new choice of the pivot (p12)(8.22)α(a,b)=a+b+d2−12(Δ2+Δ3−Δ1)β(a,b)=a+b+d−12(Δ1+Δ2+Δ3) which are P23 symmetric and(8.23)γ(a)=2a+d2−Δ2+1γ′(b)=2b+d2−Δ3+1 with P23γ(a)=γ′(b). If we require that Δ2=Δ3, as in the TJJ case, the symmetry constraints are easily implemented. Given that the 4 indices, if we choose p1 as a pivot, are given by(8.24)a0=0,b0=0,a1=Δ2−d2,b1=Δ3−d2 clearly in this case a=b and γ(a)=γ(b). F4 has the symmetry(8.25)F4(α,β;γ,γ′;x,y)=F4(α,β;γ′,γ;y,x), and this reflects in the Bose symmetry of A1 if we impose the constraint(8.26)c(1)(a1,b0)=c(1)(a0,b1).8.3The solution for A2The equations for A2 are inhomogeneous. In this case the solution can be identified using some properties of the hypergeometric differential operators Ki, appropriately splitted. We recall that in this case they are(8.27)K21A2=2A1(8.28)K31A2=2A1. We take an ansatz of the form(8.29)A2(p1,p2,p3)=p1Δ−2d−2F(x,y) which provides the correct scaling dimensions for A2. Observe that the action of K21 and K3 on A2 can be rearranged as follows(8.30)K21A2=4xaybp1Δ−2d−4(K¯21F(x,y)+∂∂xF(x,y))(8.31)K31A2=4xaybp1Δ−2d−4(K¯31F(x,y)+∂∂yF(x,y)) where(8.32)K¯21F(x,y)={x(1−x)∂2∂x2−y2∂2∂y2−2xy∂2∂x∂y+[(γ(a)−1)−(α(a,b)+β(a,b)+3)x]∂∂x+a(a−a1)x−(α(a,b)+β(a,b)+3)y∂∂y−(α+1)(β+1)}F(x,y), and(8.33)K¯31A2={y(1−y)∂2∂y2−x2∂2∂x2−2xy∂2∂x∂y+[(γ′(b)−1)−(α(a,b)+β(a,b)+3)y]∂∂y+b(b−b1)y−(α(a,b)+β(a,b)+3)x∂∂x−(α(a,b)+1)(β(a,b)+1)}×F(x,y). At this point observe that the hypergeometric function solution of the equation(8.34)K¯21F(x,y)=0 can be taken of the form(8.35)Φ1(2)(x,y)=p1Δ−2d−2∑a,bc1(2)(a,b,Δ→)xaybF4(α(a,b)+1,β(a,b)+1;γ(a)−1,γ′(b);x,y) with c1(2) a constant and the parameters a,b fixed at the ordinary values (ai,bj) as in the previous cases (8.8) and (8.9), in order to get rid of the 1/x and 1/y poles in the coefficients of the differential operators. The sequence of parameters in (8.35) will obviously solve the related equation(8.36)K31Φ1(2)(x,y)=0. Eq. (8.34) can be verified by observing that the sequence of parameters (α(a,b)+1,β(a,b)+1γ(a)−1) allows to define a solution of (8.33) set to zero, for an arbitrary γ′(b), since this parameter does not play any role in the solution of the corresponding equation. The sequence (α(a,b)+1,β(a,b)+1,γ′(b)), on the other hand, solves the homogeneous equations associated to K31 (i.e. Eq. (8.36)) for any value of the third parameter of F4, which in this case takes the value γ(a)−1. A similar result holds for the mirror solution(8.37)Φ2(2)(x,y)=p1Δ−2d−2∑a,bc2(2)(a,b,Δ→)xaybF4(α(a,b)+1,β(a,b)+1;γ(a),γ′(b)−1;x,y) which satisfies(8.38)K¯31Φ2(2)(x,y)=0K21Φ2(2)(x,y)=0. As previously remarked, the values of the exponents a and b remain the same for any equation involving either a Ki,j or a K¯ij, as can be explicitly verified. This implies that the fundamental solutions of the conformal equations are essentially the 4 functions of the type S1,…S4, for appropriate values of their parameters.At this point, to show that F1 and F2 is a solution of Eqs. (8.27) we use the property(8.39)∂p+qF4(α,β;γ1,γ2;x,y)∂xp∂yq=(α,p+q)(β,p+q)(γ1,p)(γ2,q)F4(α+p+q,β+p+q;γ1+p;γ2+q;x,y) which gives (for generic parameters α,β,γ1,γ2)(8.40)∂F4(α,β;γ1,γ2;x,y)∂x=αβγ1F4(α+1,β+1,γ1+1,γ2,x,y)∂F4(α,β;γ1,γ2;x,y)∂y=αβγ2F4(α+1,β+1,γ1,γ2+1,x,y). Obviously, such relations are valid whatever dependence the four parameters α,β,γ1,γ2 may have on the Fuchsian exponents (ai,bj). The actions of K21 and K31 on the Φ2(i)'s (i=1,2) in (8.37) are then given by(8.41)K21Φ1(2)(x,y)=4p1Δ−2d−4∑a,bc(2)1(a,b,Δ→)xayb∂∂x×F4(α(a.b)+1,β(a,b)+1;γ(a)−1,γ′(b);x,y)=4p1Δ−2d−4∑a,bc1(2)(a,b,Δ→)xayb(α(a,b)+1)(β(a,b)+1)(γ(a)−1)×F4(α(a,b)+2,β(a,b)+2;γ(a),γ′(b);x,y)K31Φ1(2)(x,y)=0(8.42)K31Φ2(2)(x,y)=4p1Δ−2d−4∑a,bc2(2)(a,b,Δ→)xayb∂∂y×F4(α(a,b)+1,β(a,b)+1;γ(a),γ′(b)−1;x,y)=4p1Δ−2d−4∑a,bc2(2)(a,b,Δ→)xayb(α(a,b)+1)(β(a,b)+1)(γ′(b)−1)×F4(α(a,b)+2,β(a,b)+2;γ(a),γ′(b);x,y)K21Φ2(2)(x,y)=0, where it is clear that the non-zero right-hand-side of both equations are proportional to the form factor A1 given in (8.21). Once this particular solution is determined, Eq. (8.21), by comparison, gives the conditions on c1(2) and c1(2) as(8.43)c1(2)(a,b,Δ→)=γ(a)−12(α(a,b)+1)(β(a,b)+1)c(1)(a,b,Δ→),(8.44)c2(2)(a,b,Δ→)=γ′(b)−12(α(a,b)+1)(β(a,b)+1)c(1)(a,b,Δ→). Therefore, the general solution for A2 in the TJJ case (in which γ(a)=γ′(b)) is given by superposing the solution of the homogeneous form of (8.21) and the particular one (8.35) and (8.37), by choosing the constants appropriately using (8.44). Its explicit form is written as(8.45)A2=p1Δ−2d−2∑abxayb[c(2)(a,b,Δ→)F4(α(a,b)+1,β(a,b)+1;γ(a),γ′(b);x,y)+(γ(a)−1)c(1)(a,b,Δ→)2(α(a,b)+1)(β(a,b)+1)(F4(α(a,b)+1,β(a,b)+1;γ(a)−1,γ′(b);x,y)+F4(α(a,b)+1,β(a,b)+1;γ(a),γ′(b)−1;x,y))], since γ(a)=γ′(b).8.4The solution for A3Using a similar strategy, the particular solution for the form factor A3 of the equations(8.46)K21A3=0K31A3=−4A1 can be found in the form(8.47)Φ(3)(x,y)=p1Δ−2d−2∑abc1(3)(a,b,Δ→)xaybF4(α(a,b)+1,β(a,b)+1;γ(a),γ′(b)−1;x,y).Also in this case the inhomogeneous equation in (8.46) fixes the integration constants to be those appearing in A1(8.48)c1(3)(a,b,Δ→)=−γ′(b)−1(α(a,b)+1)(β(a,b)+1)c(1)(a,b,Δ→). Therefore the general solution of the equations (8.46) can be written as(8.49)A3=p1Δ−2d−2∑abxayb[c(3)(a,b,Δ→)F4(α(a,b)+1,β(a,b)+1;γ(a),γ′(b);x,y)−(γ(a)−1)c(1)(a,b,Δ→)(α(a,b)+1)(β(a,b)+1)×F4(α(a,b)+1,β(a,b)+1;γ(a),γ′(b)−1;x,y)] since γ(a)=γ′(b) in the TJJ case.8.5The A4 solutionThe last pair of equations(8.50)K21A4=−2A3K31A4=−2A3(p2↔p3) admit three particular solutions(8.51)Φ1(4)=p1Δ−2d∑abxaybc1(4)(a,b,Δ→)F4(α(a,b),β(a,b),γ(a)−1,γ′(b),x,y)(8.52)Φ2(4)=p1Δ−2d∑abxaybc2(4)(a,b,Δ→)F4(α(a,b),β(a,b),γ(a),γ′(b)−1,x,y)(8.53)Φ3(4)=p1Δ−2d∑abxaybc3(4)(a,b,Δ→)F4(α(a,b),β(a,b),γ(a)−1,γ′(b)−1;x,y) with the action of K21 and K31 on them as(8.54)K21Φ1(4)=4p1Δ−2d−2∑abxaybc1(4)(a,b,Δ→)α(a,b)β(a,b)(γ(a)−1)×F4(α(a,b)+1,β(a,b)+1,γ(a),γ′(b),x,y)(8.55)K31Φ1(4)=0(8.56)K21Φ2(4)=0(8.57)K31Φ2(4)=4p1Δ−2d−2∑abxaybc2(4)(a,b,Δ→)α(a,b)β(a,b)(γ′(b)−1)×F4(α(a,b)+1,β(a,b)+1,γ(a),γ′(b),x,y)(8.58)K21Φ3(4)=4p1Δ−2d−2∑abxaybc3(4)(a,b,Δ→)α(a,b)β(a,b)(γ(a)−1)×F4(α(a,b)+1,β(a,b)+1,γ(a),γ′(b)−1;x,y)(8.59)K31Φ3(4)=4p1Δ−2d−2∑abxaybc3(4)(a,b,Δ→)α(a,b)β(a,b)(γ′(b)−1)×F4(α(a,b)+1,β(a,b)+1,γ(a)−1,γ′(b);x,y) The inhomogeneous equations (8.50) fix the integration constants to be those appearing in A3 and A3(p2↔p3) as(8.60)c1(4)=−(γ(a)−1)2α(a,b)β(a,b)c(3)(a,b,Δ→)(8.61)c2(4)=−(γ′(b)−1)2α(a,b)β(a,b)c(3)(a,b,Δ→)(8.62)c3(4)=(α(a,b)+1)(β(a,b)+1)2α(a,b)β(a,b)c(1)(a,b,Δ→). Finally, using the properties γ(a)=γ′(b), we give the general solution for the A4 as(8.63)A4=p1Δ−2d∑abxayb[c(4)(a,b,Δ→)F4(α(a,b),β(a,b),γ(a),γ′(b);x,y)+(α(a,b)+1)(β(a,b)+1)2α(a,b)β(a,b)c(1)(a,b,Δ→)×F4(α(a,b),β(a,b),γ(a)−1,γ′(b)−1;x,y)−(γ(a)−1)2α(a,b)β(a,b)c(3)(a,b,Δ→)(F4(α(a,b),β(a,b),γ(a)−1,γ′(b),x,y)+F4(α(a,b),β(a,b),γ(a),γ′(b)−1,x,y))] Notice that, differently from this case, number of free constants can be significantly reduced in the case of a fully symmetric correlator, such as the TTT, where the number of constants reduces to 4, as in the BMS case.8.6SummaryTo summarize, the solutions of the primary WI's in the TJJ case are expressed as sums of 4 hypergeometrics of universal indicial points(8.64)a0=0,b0=0,a1=Δ2−d2,b1=Δ3−d2 and parameters(8.65)α(a,b)=a+b+d2−12(Δ2+Δ3−Δ1),β(a,b)=a+b+d−12(Δ1+Δ2+Δ3)(8.66)γ(a)=2a+d2−Δ2+1,γ′(b)=2b+d2−Δ3+1, where Δ2=Δ3=d−1 and Δ1=d. In particular they are given by(8.67)A1=p1Δ−2d−4∑a,bc(1)(a,b,Δ→)xaybF4(α(a,b)+2,β(a,b)+2;γ(a),γ′(b);x,y)(8.68)A2=p1Δ−2d−2∑abxayb[c(2)(a,b,Δ→)F4(α(a,b)+1,β(a,b)+1;γ(a),γ′(b);x,y)+(γ(a)−1)c(1)(a,b,Δ→)2(α(a,b)+1)(β(a,b)+1)(F4(α(a,b)+1,β(a,b)+1;γ(a)−1,γ′(b);x,y)+F4(α(a,b)+1,β(a,b)+1;γ(a),γ′(b)−1;x,y))](8.69)A3=p1Δ−2d−2∑abxayb[c(3)(a,b,Δ→)F4(α(a,b)+1,β(a,b)+1;γ(a),γ′(b);x,y)−(γ(a)−1)c(1)(a,b,Δ→)(α(a,b)+1)(β(a,b)+1)×F4(α(a,b)+1,β(a,b)+1;γ(a),γ′(b)−1;x,y)](8.70)A4=p1Δ−2d∑abxayb[c(4)(a,b,Δ→)F4(α(a,b),β(a,b),γ(a),γ′(b);x,y)+(α(a,b)+1)(β(a,b)+1)2α(a,b)β(a,b)c(1)(a,b,Δ→)×F4(α(a,b),β(a,b),γ(a)−1,γ′(b)−1;x,y)−(γ(a)−1)2α(a,b)β(a,b)c(3)(a,b,Δ→)(F4(α(a,b),β(a,b),γ(a)−1,γ′(b),x,y)+F4(α(a,b),β(a,b),γ(a),γ′(b)−1,x,y))] in terms of the constants c(i)(a,b) given above. The method has the advantage of being generalizable to higher point functions, in the search of specific solutions of the corresponding correlation functions.9Perturbative analysis in the conformal case: QED and scalar QEDIn this section we turn to discuss the connection between the solutions of the CWI's presented by BMS and the perturbative TJJ vertex. The QED case has been previously studied in [4,7], where more details can be found. The expressions of the form factors had been given in the F-basis of 13 form factors, which will be reviewed in the next section. We will have to recompute them in order to present them expressed in terms of the two basic fundamental master integrals B0 and C0 of the tensor reduction rather than in their final form, given in [7].Here we are also going to introduce the diagrammatic expansion for the TJJ in scalar QED, since it will be needed in the last part of the work when we are going to compare the general BMS solution against the perturbative one in d=3 and d=5.The quantum actions for the fermion field is(9.1)Sfermion=i2∫ddxeeaμ[ψ¯γa(Dμψ)−(Dμψ¯)γaψ], eaμ is the Vielbein and e its determinant, with its covariant derivative Dμ as(9.2)Dμ=∂μ+ieAμ+Γμ=∂μ+ieAμ+12Σabeaσ∇μebσ. The Σab are the generators of the Lorentz group in the case of a spin 1/2-field. The gravitational field is expanded, as usual, in the form gμν=ημν+hμν around the flat background metric with fluctuations hμν. As usual, the Latin and Greek indices are related to the locally flat and curved backgrounds respectively. We take the external momenta as incoming. In order to simplify the notation, we introduce the tensor components(9.3)Aμ1ν1μν≡ημ1ν1ημν−2ημ(μ1ην1)ν where we indicate with the round brackets the symmetrization of the indices and the square brackets their anti-symmetrization(9.4)ημ(μ1ην1)ν≡12(ημμ1ην1ν+ημν1ημ1ν) and the vertices in the fermion sector are(9.5)VJψψ¯μ(k1,k2)=−ieγμ(9.6)VTψψ¯μ1ν1(p1k1,k2)=−i4Aμ1ν1μνγν(k1+k2)μ(9.7)VTJψψ¯μ1ν1μ2(k1,k2)=ie2Aμ1ν1μ2νγμ.In the one-loop approximation the contribution to the correlation functions are given by the diagrams in Fig. 1, with vertices shown in Fig. 2. We calculate all the diagram contributions in momentum space for the fermion sector as(9.8)Γμ1ν1μ2μ3(p2,p3)≡〈Tμ1ν1(p1)Jμ2(p2)Jμ3(p3)〉F=2(∑i=12VF,iμ1ν1μ2μ3(p1,p2,p3)+∑i=12WF,iμ1ν1μ2μ3(p1,p2,p3)) where the VF,i terms are related to the triangle topology contributions, while the WF,i terms denote the two bubble contributions in Fig. 1. All these terms are explicitly given as(9.9)VF,1μ1ν1μ2μ3=−i3∫ddℓ(2π)d×Tr[VTψψ¯μ1ν1(ℓ−p2,ℓ+p3)(ℓ̸+p̸3)VJψψ¯μ2(ℓ,ℓ−p2)ℓ̸VJψψ¯μ3(ℓ,ℓ+p3)(ℓ̸−p̸2)]ℓ2(ℓ−p2)2(ℓ+p3)2(9.10)VF,2μ1ν1μ2μ3=−i3∫ddℓ(2π)d×Tr[VTψψ¯μ1ν1(ℓ−p3,ℓ+p2)(ℓ̸+p̸2)VJψψ¯μ2(ℓ,ℓ−p3)ℓ̸VJψψ¯μ3(ℓ,ℓ+p2)(ℓ̸−p̸3)]ℓ2(ℓ−p3)2(ℓ+p2)2(9.11)WF,3μ1ν1μ2μ3=−i2∫ddℓ(2π)dTr[VTJψψ¯μ1ν1μ2(ℓ+p3,ℓ)(ℓ̸+p̸3)VJψψ¯μ3(ℓ,ℓ+p3)ℓ̸]ℓ2(ℓ+p3)2(9.12)WF,2μ1ν1μ2μ3=−i2∫ddℓ(2π)dTr[VTJψψ¯μ1ν1μ3(ℓ+p2,ℓ)(ℓ̸+p̸2)VJψψ¯μ2(ℓ,ℓ+p2)ℓ̸]ℓ2(ℓ+p2)29.1The TJJ in scalar QEDNow we turn to consider scalar QED. The action, in this case, can be written as(9.13)Sscalar=∫ddx−g(|Dμϕ|2+(d−2)8(d−1)R|ϕ|2) where R is the scalar curvature and ϕ denotes a complex scalar. We have explicitly reported the coefficient of the term of improvement, and with Dμϕ=∂μϕ+ieAμ being the covariant derivative for the coupling to the gauge field Aμ. At one-loop the contribution to the TJJ is given by the diagram in Fig. 1, with the obvious replacement of a fermion by a scalar in the internal loop corrections. In this case they are given by(9.14)〈Tμ1ν1(p1)Jμ2(p2)Jμ3(p3)〉S=2(VSμ1ν1μ2μ3(p1,p2,p3)+∑i=13WS,iμ1ν1μ2μ3(p1,p2,p3)) where the VS terms are related to the triangle topology contribution and the WS,i's are the three bubble contributions in Fig. 1. All these are explicitly given as(9.15)VSμ1ν1μ2μ3(p1,p2,p3)=i3∫ddℓ(2π)dVTϕϕ⁎μ1ν1(ℓ−p2,ℓ+p3)VJϕϕ⁎μ2(ℓ,ℓ−p2)VJϕϕ⁎μ3(ℓ,ℓ+p3)ℓ2(ℓ−p2)2(ℓ+p3)2(9.16)WS,1μ1ν1μ2μ3(p1,p2,p3)=i22∫ddℓ(2π)dVTϕϕ⁎μ1ν1(ℓ+p1,ℓ)VJJϕϕ⁎μ2μ3(ℓ,ℓ+p1)ℓ2(ℓ+p1)2(9.17)WS,2μ1ν1μ2μ3(p1,p2,p3)=i22∫ddℓ(2π)dVμ1ν1μ2TJϕϕ⁎(ℓ+p3,ℓ)VJϕϕ⁎μ3(ℓ,ℓ+p3)ℓ2(ℓ+p3)2(9.18)WS,3μ1ν1μ2μ3(p1,p2,p3)=i22∫ddℓ(2π)dVμ1ν1μ3TJϕϕ⁎(ℓ+p2,ℓ)VJϕ