Higher spin 3-point functions in 3d CFT using spinor-helicity variables

In this paper we use the spinor-helicity formalism to calculate 3-point functions involving scalar operators and spin-$s$ conserved currents in general 3d CFTs. In spinor-helicity variables we notice that the parity-even and the parity-odd parts of a correlator are related. Upon converting spinor-helicity answers to momentum space, we show that correlators involving spin-$s$ currents can be expressed in terms of some simple conformally invariant conserved structures. This in particular allows us to understand and separate out contact terms systematically, especially for the parity-odd case. We also reproduce some of the correlators using weight-shifting operators.

3d CFT correlators for conserved currents in position space are quite well explored. A detailed position space analysis of higher spin CF T 3 correlators was performed in [61]. For the position space 3-point correlators of conserved currents of the form J s 1 J s 2 J s 3 , where s i ≥ 1, it was shown that the correlators have two parity-even structures and one parity-odd structure. When the sum of spins is even, the parity-even contributions arise from three-point correlators in the free-boson and free-fermion theories. When the sum of spins is odd, the parity-even contributions arise from correlators of non-abelian currents in the free theory of multiple scalars or fermions. The parity-odd structure is not generated by a free theory. When the spins of the conserved currents satisfy the triangle inequality, i.e. s i ≤ s i+1 + s i+2 for i = 1, 2, 3 (modulo 3), there exists one parity-odd structure. When the inequality is not satisfied the odd structure is zero. A proof of this using an integral representation of the correlators was presented in [62]. It was also noted in [61,62] that a correlator of the form J s J s O ∆ , where O ∆ is a scalar operator with scaling dimension ∆, contains only two structures, one parity-odd and another parity-even. Also, a correlator of the form J s O ∆ 1 O ∆ 2 is non-zero only for ∆ 1 = ∆ 2 and only has one parity-even structure. In [61] it was shown that in position space, one can write down arbitrary CFT 3-point higher spin correlators as multinomials of just a few simple conformal invariants.
A similar exhaustive analysis of CFT correlators in momentum space or spinor-helicity variables has not been done yet. A helicity basis was used in [63] for studying higher spin 3d CFT correlators. Spinor-helicity variables have been used earlier in the study of CFTs, see for example [3,7,12]. Our goal in this paper is to study 3-point correlators of scalar operators and conserved spin-s currents in d = 3 CFTs in spinor-helicity variables as well as in momentum space.
In [2], parity-even 3-point momentum space correlators involving scalar operator and conserved currents up to spin-2 (stress tensor) were computed by solving the conformal Ward identities (CWI). These correlators were later obtained using weight-shifting operators in [11,12]. In [50], we explored momentum space CF T 3 parity-odd 3-point correlators such as JJO ∆ by solving CWIs directly in momentum space. We also calculated parity-odd correlators of the form T T O ∆ and JJJ using weight-shifting operators, as the direct use of CWIs became very complex and inefficient. More complex higher spin correlators were not computed directly using momentum space CWIs 1 .
The main difficulty in calculating complicated correlators such as T T T is that there is a high degree of degeneracy in the tensor structures in 3d, both in the parity-even and the parity-odd sector, which makes it difficult to choose an appropriate basis to write an ansatz for the correlator. In the parity-odd sector one has to deal with the additional complication of non-trivial Schouten identities, which makes it difficult to solve for T T T odd directly. The problem becomes even more complicated if we want to calculate a correlator involving higher spin conserved currents (J s with s > 2) both for the parity-even and parity-odd case. In this paper, we overcome this problem by working in the spinor-helicity formalism where the degeneracy is automatically taken care of. We solve the CWIs in these variables and then convert the results back to momentum space. In this way we obtain the momentum space expressions for all correlators of the form J s 1 J s 2 J s 3 with spins satisfying the triangle inequality. We make contact with the counting of structures in [61] by showing that correlators of conserved spinning operators in momentum space have 2 parity-even and 1 parity-odd structure up to contact terms.
In our analysis we split the correlation function into two pieces, homogeneous and non-homogeneous parts. We show that for the correlation functions we consider the parityodd contribution to the non-homogeneous piece is always a contact term. Interestingly, spinor-helicity variables reveal that parity-even and parity-odd contributions to homogeneous pieces are completely identical, although they look completely different in momentum space as well as position space. Moreover, for divergent correlation functions which require completely different regularization and renormalization for parity-even and parityodd parts, the relation between these parts in spinor-helicity variables holds even after renormalization. Upon converting spinor-helicity answers to momentum space, we see that the results for correlators involving spin-s currents can be expressed in terms of some simple conformally invariant conserved structures. In certain cases (such as T T O 4 ), the correlators are divergent in momentum space and require a careful application of the renormalization procedure, but in spinor-helicity variables they turn out to give directly the finite part without any renormalization. We also verify some of the results using weight-shifting operators.
The plan of the rest of the paper is as follows. In Section 2 we introduce the basic idea of expressing conformal correlators in terms of spinor-helicity variables and discuss the preliminary case of 2-point functions. We also discuss some general features of our 3point function analysis. Section 3 has the results of various 3-point correlators of spinning conserved currents and scalar operators in spinor-helicity variables. In Section 4 we translate these results to momentum space after carefully taking the degeneracies into account and Section 5 has a discussion of the renormalisation of some of these correlators which have divergences. In Section 6 some of these momentum space results are re-derived using weight-shifting operators acting on seed correlators. Section 7 contains a discussion of momentum space higher-spin conserved current correlators expressed in terms of 3-point momentum space invariants. In Section 8 we make some important observations, including the connection between the parity-even and parity-odd parts of a correlator. We conclude in Section 9 with a brief summary and a discussion on future directions of study. At the end we have a number of appendices supplementing the main text and providing various technical details. Appendix A outlines our spinor-helicity notation. In Appendix B we describe in detail our terminology of homogeneous and non-homogeneous contributions to a correlator and discuss how they differ from the usual splitting of a correlation function into transverse and longitudinal pieces. Appendix C has the technical details of solutions of various conformal Ward identities quoted in Section 3. Appendix D contains useful triple-K integral identities and Appendix E lists the momentum space form of various 3-point correlators of conserved currents. Finally Appendix F contains the required details of some weight-shifting operators which are used in Section 6.

Conformal correlators in spinor-helicity variables
Momentum space expressions for parity preserving two and three-point conformal correlators of spinning operators were obtained in [1,2,[19][20][21][22] by solving momentum space conformal Ward identities. Recently in [50] we derived momentum space expressions for parity-odd correlators using two different techniques. The first one, following [2], involved solving conformal Ward identities directly in momentum space. The second one, following [11], involved using the technique of spin-raising and weight-shifting operators in momentum space. In [42,43] following the position space analysis in [62,64] it was shown that one could make use of momentum space higher-spin equations arising from Ward identities associated to (weakly broken) higher spin symmetry to compute spinning correlators including the parity-odd ones.
The analysis of parity-odd correlators was restricted to correlators such as JJO , T T O and JJJ due to various technical difficulties. One of the main obstacles was to identify the correct basis of tensor structures to work with, due to various non-trivial Schouten identities and other degeneracies in three-dimensions.
In this section, we compute 3-point CFT correlators in spinor-helicity variables. It turns out that solving for CFT correlators in spinor-helicity variables is a lot simpler than doing so in momentum space. The reader may wish to refer Appendix A at this point to get familiar with our notation and convention regarding spinor-helicity variables.
We start with an ansatz for the correlator in spinor-helicity variables. To do so, we use the fact that a Lorentz transformation of the momentum k corresponds to a scale transformation of the spinors. Therefore, a Lorentz-covariant structure in spinor-helicity variables is a structure that has the correct scaling based on the helicities of the operators. An operator O with helicity h transforms in the following way under a scale transformation of spinors : Therefore, the ansatz for a general correlator is given by where F 1 (k 1 , k 2 , k 3 ) and F 2 (k 1 , k 2 , k 3 ) are form-factors that we will determine by imposing dilatation and special conformal invariance. For parity-even correlators c 2 = 0 and for parity-odd correlators c 1 = 0, and for the latter the 'i' ensures that the correlator changes sign under conjugation, since conjugation corresponds to a parity transformation for spinors.

Conformal generators
The conformal Ward identities are differential equations determined by the action of the special conformal generator on a conformal correlator. The special conformal generator in spinor-helicity variables takes the form [65] : 3) The action of K on a scalar with ∆ = 2 is given by [12] : The action of K on a scalar with ∆ = 2 is given by [12] : Similarly, the action of K on spin-one and spin-two conserved currents is as follows [12] : where J + = z + µ J µ and T + = z + µ z + ν T µν . In (2.5) and (2.7), K κ corresponds to the special conformal generator in momentum space with ∆ = 2. Its action on a conformally invariant correlator is zero. Therefore, the action of K κ on a correlator in which all the operators have ∆ = 2 will just have a part proportional to the R.H.S. of the Ward-Takahashi identity of the correlator. When the correlator has operators with scaling dimensions other than 2, it is convenient to divide them by appropriate powers of k so that the insertion has ∆ = 2. For a derivation, see [3].

Three-point functions: General discussion
We will now consider three-point functions with spinning operator insertions. The parity odd sector of a few correlators such as JJO , JJJ , and T T O have been studied in momentum space by solving conformal Ward identities, using spin-raising and weight-shifting operators and using higher spin equations [42,43,50]. In extending our analysis to more complicated three-point correlators we faced some difficulties as described in the beginning of this section. However, working in spinor-helicity variables, we are able to circumvent this problem and get expressions for more complicated 3-point correlators as described in detail below. We will first introduce the terminology of homogeneous and non-homogeneous solutions to conformal Ward identities which we will use throughout this paper.
= transverse Ward identity terms (2.12) This distinction will be important to keep in mind since the homogeneous and non-homogeneous parts have different structures and properties. One way to distinguish between the two kinds of solutions in the final answer will be that the non-homogeneous solution depends on the coefficient of the two-point function. Another way is to make use of the transverse Ward identities : In other words, while the homogeneous solution is completely transverse, the non-homogeneous solution gets contribution from both transverse as well as local (or longitudinal) terms.
Since the 3-point correlators can be parity-violating, it will be useful to break up the homogeneous and non-homogeneous parts further into parity-even and parity-odd contributions: For a detailed discussion on the homogeneous and non-homogeneous contributions to threepoint correlators and their distinction from transverse and longitudinal contributions see appendix B.

Degeneracy structure
In three dimensions, there exist degeneracies in tensor structures which complicate the analysis of correlators. The existence of degeneracy is tied to the simple fact that not more than three vectors can be linearly independent in three dimensions. The basic problem is that the different tensor structures in the ansatz for a correlator become linearly dependent due to degeneracies. This affects the analysis of both parityeven and parity-odd correlators. For the parity-odd correlator, Schouten identities, which relate various tensor structures involving Levi-Civita tensors, are an additional source of complication. The main problem is that while solving the conformal Ward identity, one needs to identify the correct independent set of tensor structures to be able to write down differential equations for the form-factors. However, this process becomes very complicated for correlators involving spin-2 or higher spin operators.
An example of such an identity in three dimensions is : where we have used the notation 2 z 2 k 1 k 2 = µνρ z µ 2 k ν 1 k ρ 2 . The structures that appear in the above equation arise in the ansatz for various parity-odd correlators such as JJO odd . The above equation then implies that a term with z 1 k 1 k 2 in the ansatz can be eliminated in favour of other structures 3 . This, while essential to be taken into account, makes cumbersome the correct ansatz with a minimal basis of independent structures.
Other than Schouten identities, there are identities such as [2] : where n µ = µνρ k ν k ρ and i = j = 1, 2, 3. We also have [2] : Another example of a degeneracy is [20] : These also allow certain basis structures to be expressed in terms of others. Both parity-even and parity-odd degeneracies complicate the analysis when computing correlation functions.
One of the advantages of working with spinor-helicity variables is that the degeneracies become trivial in these variables. For example, the left hand side of both (2.15) and (2.18) become identically zero in spinor-helicity variables. One can check that all the Schouten identities and other identities relating various tensor structures also become trivial in spinorhelicity variables.

Three-point functions : Explicit solutions in spinor-helicity variables
In this section we focus on determining CF T 3 3-point correlators in spinor-helicity variables. In particular, we compute correlators of the form J s O ∆ O ∆ , J s J s O ∆ , J s J s J s and J s 1 J s J s where J s is a symmetric, traceless, spin-s conserved current with scaling dimension ∆ = s + 1, and O ∆ is a scalar operator with scaling dimension ∆. In three dimensions, 3-point correlators involving only spinning operators are always finite, whereas those involving a scalar operator require renormalization for large enough values of ∆.
We will observe that splitting the correlator into homogeneous and non-homogeneous parts in the sense explained in Section 2.3.1 is useful. As we demonstrate, whenever there exists a homogeneous parity-even solution to the conformal Ward identity in spinor-helicity variables, there also exists a homogeneous parity-odd solution and the two are identical up-to some signs. Interestingly, in the case of divergent correlators, the parity-odd and the parity-even correlators continue to match even after renormalization, although the renormalization procedure for the two differs. Further, it turns out that the non-homogeneous part is always parity-even. Any parity-odd contribution to the non-homogeneous part is always a contact term. After the first example in which we present all the details, in each case we will give the correlator ansatz and then write down the form-factors as solution of the CWI's, relegating the details to Appendix C. 3.1 In this section, we calculate correlators of the form J s O ∆ O ∆ . The Ward-Takahashi (WT) identity when the spinning operator is either a spin-one conserved current or the stresstensor (i.e. when s = 1 or s = 2) is given by the following [2,12]: where in the second equation we have contracted both sides of the WT identity with null transverse polarization vectors. It is straightforward to generalise the WT identity to arbitrary spin-s conserved currents by matching the spin and scaling dimensions on both sides of the identity. This gives the following : We will see that the homogeneous part of the correlator is zero. The non-homogeneous part has the scalar two-point function on the right hand side 4 . Consequently, the odd part of the correlator goes to zero as there is no parity-odd scalar two-point function. Thus this correlator has only a parity-even non-homogeneous part. As noted in Section 2.1, when the correlator involves operators with scaling dimensions other than 2, it is convenient to divide the insertions by appropriate powers of the corresponding momenta k such that they have ∆ = 2. The correlator itself is obtained at the end by restoring the powers of k. Keeping this in mind, we start with the following ansatz for the correlator : The action of the generator of special conformal transformations K is then given by (see Section 2.1) : A correlator comprising one conserved current and two scalar operators with different scaling dimensions also vanishes, i.e.
gives the following : Finally, the dilatation Ward identity is given by : The above differential equations (3.6),(3.7), (3.8) and (3.9) can be solved to get : where the triple-K integral [2] which occurs in the RHS of this equation is defined in (5.1). After taking the momentum factors in the denominator of the LHS of (3.4) to the RHS and using the above result for the form factor we obtain the correlator : When c O = 0, there is no non-trivial solution to the differential equations and one has : In this section, we compute correlators of the form J s J s O ∆ for general spin s. As discussed in Section 2.3.1, we separate out the correlator into homogeneous and non-homogeneous parts : The correlator J s J s O ∆ is completely transverse : where k · J s (k) = k µ 1 J µ 1 µ 2 ....µs (k). This implies that the non-homogeneous part of the correlator is zero : We will now compute the explicit form of the correlators for arbitrary ∆. We find that for ∆ ≥ 4, there is a divergence and we need to regularize and renormalize to obtain finite correlators. We consider the following ansatz for the correlator : (2.2) : Substituting the form-factor in the ansatz (3.16) we obtain For ∆ ≥ 4, the above triple-K integrals and thereby the correlators are divergent. A detailed study of the renormalization of these correlators will be carried out in Section 5.
We will see that the relationship between the parity-even and the parity-odd parts of a correlator in spinor-helicity variables continues to hold even after renormalization.

J s J s J s
In this subsection we concentrate on the three point function of a general spin s conserved current J s 5 . Since the correlator J s J s J s satisfies a nontrivial transverse WT identity it has both the homogeneous as well as the non-homogeneous contributions. Let us split the correlator into the odd and even contributions : It will turn out that J s J s J s even has both the homogeneous and the non-homogeneous contributions whereas J s J s J s odd has a non-trivial homogeneous part but the non-homogeneous part is always a contact term.

JJJ
Let us start our analysis with the 3-point function of the spin-1 current J µ . As noted earlier, for this correlator to be non-zero, the currents have to be non-abelian. The WT identity is given by [2,12,20] : Let us consider the following ansatz for the two helicity components of the correlator 6 : The solutions of the conformal Ward identity are given by (see Appendix C.3) where c J and c J are the parity-even and parity-odd coefficients of the two-point function of conserved currents (see (2.8)). The terms proportional to c 1 , c 1 and c 2 , c 2 are the homogeneous solutions to the differential equations and those proportional to c J , c J are the non-homogeneous solutions. Since G(k 1 , k 2 , k 3 ) and G(k 1 , k 2 , k 3 ) both have an un-physical pole when k 1 + k 2 = k 3 , we set the coefficients of these terms to zero, i.e. c 2 = c 2 = 0.

Summary of the solution
Taking into account both the parity-even and the parity-odd contributions, we obtain : In the next section, we will convert these expressions into the momentum space and see that the non-homogeneous contribution to the parity-odd correlator (term proportional to c J ) becomes a contact term.

T T T
Let us now consider the correlator with three insertions of the stress-tensor operator. The transverse Ward identity satisfied by the correlator is given by [2,12,20]: Thus the correlator can have both homogeneous and nonhomogeneous solutions for the parity-even and parity-odd correlation functions. The parityeven solution was already discussed in [2]. The parity-odd homogeneous contribution was computed in [7] using Feynman diagram computations in dS 4 . Here we reproduce the same result by solving the conformal Ward identities. We also get a nontrivial non-homogeneous contribution to the parity-odd correlator which in momentum space turns out to be a contact term.

T T T even
The parity-even contribution to the correlator T T T even is given by [2,12] where b 123 = (k 1 k 2 + k 2 k 3 + k 3 k 1 ), c 123 = k 1 k 2 k 3 , and c T comes from the parity-even two point function of the stress tensor (2.8). The terms proportional to c 1 and c T are the homogeneous and the non-homogeneous contributions to the correlator respectively.

T T T odd
The parity-odd part of the correlator can be solved analogously (see Appendix C.4 for details). The answer is given by where a 12 = k 1 + k 2 , b 123 = k 1 k 2 + k 2 k 3 + k 1 k 3 and c 123 = k 1 k 2 k 3 and c T arises in the parity-odd two point function of the stress tensor (2.8). The terms proportional to c 1 and c T are the homogeneous and the non-homogeneous contributions to the correlator respectively.

Summary of the solution
Taking into account both the parity-even and the parity-odd contributions, we obtain : The other helicity components of the correlator can be obtained by complex conjugating the above results.
In the next section, we will show that the non-homogeneous contribution to the parity odd correlator (the term proportional to c T in (3.32)) is a contact term. Thus the only non-trivial contribution to the non-homogeneous piece in the correlator comes from the parity-even part.

3.3.3
J s J s J s for general spin: The homogeneous part In this subsection we generalize the above discussion to the three-point function J s J s J s of arbitrary spin s conserved current. We first split the correlator into homogeneous and non-homogeneous pieces, and then separate them into their parity-even and parity-odd parts as indicated in (2.14). The non-homogeneous piece J s J s J s nh requires us to know the WT identity which for general spin is quite complicated. However, on general grounds we can argue that the parity-odd part of this term, J s J s J s nh,odd , is a contact term for general spin s. We refer the reader to Section G for details. In the rest of this subsection, we focus on obtaining the homogeneous part of the correlator which does not require the WT identity.

Homogeneous solution
For the homogeneous solution, the parity-even and the parity-odd contributions are again the same in spinor-helicity variables. We start with the following ansatz for J s J s J s h : Since we are focusing on the homogeneous part, the action of the conformal generator is given by : The action of K κ on the ansatz, after dotting with The dilatation Ward identity is given by : The solutions for F and G are then given (see appendix C) by

Summary of result
Considering the parity-even and the parity-odd contributions we obtain the homogeneous part of the correlator : The other helicity components can be obtained by a simple complex conjugation.
In this subsection we generalize the above discussion to three-point correlators of the kind J s 1 J s J s for arbitrary spins s and s 1 . We again find it convenient to split the correlator into various parts as in (2.14). The WT identity for J s 1 J s J s for general spins s and s 1 is quite complicated. However, as discussed in Section G, we can argue that the parityodd contribution to the non-homogeneous part, J s J s J s nh,odd , is a contact term. In the following we will calculate the homogeneous and the non-homogeneous contribution to the correlator T JJ . For general spins s and s 1 , we present only the homogeneous solution.

T JJ
Let us consider the correlator with a single insertion of the stress-tensor operator and two insertions of the spin-one current operator. The transverse WT identity is given by : [2,20] Since the WT identity is not trivial, the correlator can have both homogeneous and nonhomogeneous solutions for the parity-even and the parity-odd correlation functions. The parity-even solution was already discussed in [2,20].

T JJ even
The even part of the correlator was calculated in [2,20] in momentum space and it is straightforward to convert that into spinor-helicity variables : where the term proportional to c 1 is the homogeneous term and the term proportional to c J is the non-homogeneous term.

T JJ odd
Let us now consider the parity-odd contribution to the correlator. In the parity-odd case the transverse WT identity (G.8) gives : Interestingly it turns out that the R.H.S. of the above equation vanishes upon using a Schouten identity. Thus, in addition to the trivial transverse WT identities w.r.t k ρ 2 and k σ 3 , we have the following trivial transverse WT identity : This immediately implies that the parity odd part of the non-homogeneous part of the correlator is zero : We now turn our attention to computing the homogeneous contribution. Let us start with the following ansatz for T JJ odd : The solutions to the resulting CWIs are given (see appendix C) by : Since the solution for G has an unphysical pole, we set its coefficient c 2 = 0. Substituting the above form-factors in the ansatz (3.48), we obtain the following: The other helicity components of the correlator are zero.

Summary of Homogeneous contribution to T JJ
Adding up the contribution coming from the parity-even and parity-odd parts we obtain with all other components being zero.

Summary of non-homogeneous contribution to T JJ
As discussed above, the parity-odd contribution to the non-homogeneous part of the correlator vanishes. Thus from (3.44) we have the following for the non-homogeneous part of the correlator : The solutions for F , G and H are given (see appendix C) by: (3.57) We will now summarise the results for the homogeneous solution.

Summary of Homogeneous contribution to
while other components can be obtained by complex conjugating the result in (3.58).

Conformal correlators in momentum space
In this section we present the results for higher spin CF T 3 correlators in momentum space. As explained in Section 2.3.2 a direct computation of parity-odd correlators in momentum space becomes complicated due to the large amount of degeneracy in 3d. Rather than solving the CWIs directly in momentum space, we convert our expressions for the correlators in spinor-helicity variables obtained in the previous section to momentum space. The simplest way to do this is to write down the ansatz for the correlator in momentum space and convert it to spinor-helicity variables. One can then compare it to the explicit results in spinor-helicity variables and solve for the form factors. For correlators involving higher spins, this procedure also becomes complicated, and in such cases we make use of transverse polarization tensors to represent the answers.

Two point function
Two-point functions of various conserved currents are as follows : For arbitrary spin s, we have the following expression for the two-point function after contracting with polarization vectors : From (4.5), it is clear the the parity-odd two-point function for a spin-s current is a contact term as it is analytic in k 2 .

Three point function
The parity-even sector of momentum space 3-point correlators involving spin 1 and spin 2 conserved currents and scalar operators of arbitrary scaling dimensions was obtained by solving conformal Ward identities in [2,19,20,22]. The parity odd sector of CF T 3 was studied for specific correlators using momentum space Ward identities and weightshifting operators in [51]. In this section we convert the spinor-helicity expressions in the previous section to momentum space expressions and obtain parity-even as well as parityodd three point correlators comprising generic spin s conserved currents and scalar operators of arbitrary scaling dimensions.
In this section we give the momentum space expression for correlators of the form J s O ∆ O ∆ . As discussed earlier, the parity-odd part is zero. The parity-even part is straightforward to write down from the spinor-helicity expressions. For general spin-s, it is given by : Let us now consider the correlator for some specific values of s and ∆.
For s = 1, s = 2 and s = 3, we have For s = 2 and ∆ = 3, we have Let us now consider the three-point correlator with two insertions of the spin-s conserved current and a scalar operator of scaling dimension ∆.
In this section we determine the momentum space expression for correlators of the kind J s J s O ∆ . We first discuss the s = 1 and s = 2 cases in detail. We then present the final result for the general spin case expressed in terms of a transverse polarization tensor.

JJO ∆
The correlator is purely transverse and the even part of it takes the following form in momentum space [22]: where the form factor A(k 1 , k 2 , k 3 ) is given by : and As an example let us consider the case when the scaling dimension of the scalar operator is ∆ = 2. In this case, after evaluating the integral (4.11) we obtain the form factor in the ansatz (4.10) to take the form : Let us now consider the parity-odd sector. In [51] we computed JJO ∆ odd by imposing conformal invariance and obtained : where where I 1 and I 2 are the following two triple-K integrals respectively : We further verified our results by computing the correlator for ∆ = 1, . . . , 5 using weightshifting operators and matching with the results obtained above. Again, when ∆ = 2, after evaluating the above triple-K integrals we obtain the form factors in (4.14) to be : Note that, as expected there is a total energy singularity when E = k 1 +k 2 +k 3 → 0. It seems from the above expression that there is also an apparent collinear divergence when any two momentum vectors are proportional to each other. In this case, momentum conservation implies that all 3 momenta are along a line and it is easy to check that the ((k 1 − k 2 ) 2 − k 2 3 ) factor in the denominator above vanishes. However, in this case the numerator of the full correlator also vanishes appropriately, leaving the correlator finite. Hence as expected the correlator has no singularities other than the E → 0 singularity 7 .

T T O ∆
The transverse and traceless part of the correlator in three dimensions is given by [22]: where and χ is defined in (4.12). For the case when ∆ = 2 one obtains the following : Let us now consider the parity odd contribution to the correlator. In [50] we obtained the momentum space expressions for T T O 1 odd and T T O 2 odd using spin-raising and weightshifting operators. We will now use the expressions we obtained in spinor-helicity variables for T T O ∆ , for a generic ∆, to obtain a momentum space expression for the same. We start with the following ansatz in momentum space : Symmetry considerations tell us that : Dotting with transverse, null polarization vectors, we get Converting into spinor-helicity variables we get : Comparing (4.25) and (C.14), we get the following for the form factor : which matches the form factor (4.20) that appears in the even part of the same correlator.
Let us now come to the case where the spinning operator in the correlator is a generic spin s conserved current.
As mentioned in the introduction to this section, for correlators involving higher spin operators, it is convenient to introduce transverse polarization tensors. It is straightforward to write down the momentum space expression for these correlators from their expression in spinor-helicity variables. This can be done upon observing that We then have Let us now look at this correlator for specific values of the scaling dimension of the scalar operator.
For ∆ = 1, the correlator is given by : For ∆ = 2, the correlator is given by : For ∆ = 3, the correlator is given by : We will now determine the correlator J s J s J s in momentum space. We first discuss the s = 1 and s = 2 cases in detail. For the case of general spin, we present only the final result expressed in terms of transverse polarization tensor. For this, we restrict our attention to the homogeneous part. For the parity-odd part of the correlator, the non-homogeneous contribution is always a contact term.

JJJ
Unlike J s J s O ∆ the correlator JJJ is not purely transverse and it has a local term given by [20]: The transverse part of the even part of the correlator denoted by j µ 1 a 1 j µ 2 a 2 j µ 3 a 3 even was computed in [20] : where the form factors are given by : Here and in the following we suppress the color indices for brevity. After dotting with transverse, null polarization vectors, the correlator can be separated into homogeneous and non-homogeneous parts as follows : In [50] we computed the odd part of JJJ using the action of spin-raising and weightshifting operators on a scalar seed correlator. The correlator is given by the sum of its local terms (4.33) and transverse parts. The latter is given by : After dotting with transverse polarization vectors, the correlator can be rewritten in the following form using Schouten identities : We see that the non-homogeneous part of the parity-odd part of the correlator is a contact term. This term can be explained from the dS 4 perspective by considering the three-point tree-level amplitude arising from the interaction term F F . In the rest of this section we obtain the momentum space expressions for the correlators T JJ and T T T . As described in Section 2.3.2, a direct computation of these correlators by solving the conformal Ward identities in momentum space is quite difficult.
Our strategy to circumventing the difficulties follows our approach in Section 4.2.2 to obtain T T O ∆ . We start with an ansatz for the correlator in momentum space and then convert it into spinor-helicity variables. We then compare it with the explicit results obtained in the previous section. This gives us algebraic equations involving the momentum space form-factors which can then be solved.

T T T
The momentum space expression for the even part of the correlator T T T was obtained in [20] and it was shown to have two structures. We will now obtain the momentum space expression for the odd part of the correlator.
In (3.30) we obtained the following result for the parity odd part of the correlator T T T in spinor-helicity variables : Let us consider the following ansatz for the transverse part of the correlator : One could have started with a more general ansatz with many more tensor structures than exhibited by (4.42). However, it turns out that there are several Schouten identities that relate those tensor structures and upon using them one ends up with the minimal (and complete) ansatz in (4.42) 8 . Contracting with null and transverse polarization vectors, we get Converting this into spinor-helicity variables, we get Comparing (4.41) and (4.44) and solving for the momentum space form factors we get : Naively it might look like there are two contributions to the parity-odd correlation function. However, as will be shown below the term proportional to c T is a contact term.

Contact term in parity-odd T T T
The fact that the term proportional to c T is a contact term can be seen more explicitly by switching to a basis where the factor of J 4 in the denominator disappears. One such basis is given by 9 : (4.47) The ansatz in (G. 19) is related to (4.43) by Schouten identities. Converting (G.19) into spinor-helicity variables and comparing with (4.44), we can solve for B 1 and B 2 . We get the solutions for the terms proportional to c T to be : The fact that B 1 is a constant and B 2 is dependent on k 2 implies that if we evaluate T T T in the basis (G. 19) and convert it to position space, we will get delta functions or derivatives on delta functions which are nothing but contact terms. Since the odd non-homogeneous part is a contact term, the full T T T correlator has only 3 non-trivial contributions, 2 parity-even and 1 parity-odd conformally invariant structures. This agrees with the analysis of [62].
From the dS 4 perspective this contribution to the correlator can be understood to arise from the W W interaction. The W W interaction also reproduces the parity-odd two-point function of the stress tensor.
For general spin s it is easy to write down the homogeneous part of J s J s J s in momentum space using the transverse polarization. The answer is given by : The parity-odd contribution to the non-homogeneous piece is just a contact term.

J 2s J s J s
We will now look at correlators of the form J 2s J s J s . We focus the discussion on the T JJ correlator and also give the results for the J 4 T T correlator.

T JJ
We saw in (G.8) and (G.9) that the odd part of the correlator T JJ satisfies trivial transverse Ward identities 10 . We also note that in three-dimensions the trace Ward identity obeyed by this correlator is trivial. Taking these into account we write down the following ansatz for the correlator in momentum space : Let us now contract this with polarization vectors and this gives : 10 As a result the non-homogeneous part of this parity-odd correlator is zero. One can understand this from the dS4 perspective by the following argument. The only interaction term that could possibly have contributed to this correlator is F F . However since this term is independent of the metric the contribution from it to T JJ is zero. In fact there is no interaction term that one can have from the gravity side that contributes to the non-homogeneous parity-odd part of T JJ .
We now convert this expression into spinor-helicity variables to obtain : We obtained the following explicit results for these correlators in (3.50) : Comparing (4.53) and (4.52), we get the following solutions for the form factors : Plugging back the solution (4.54) in (4.51) we obtain : which matches the result in [51] obtained by computing a tree level dS 4 amplitude.
The solution (4.55) is not manifestly symmetric under a (2 ↔ 3) exchange in this basis. However, we can use Schouten identities to convert the ansatz (4.51) to the following form where it is manifestly symmetric under a (2 ↔ 3) exchange : The relation between the form-factors in the two bases (4.51) and (4.56) is given by : For the case s 1 = 4 and s = 2, the momentum space expression that we get after converting the answer in spinor-helicity variables given in Section 3.4.2 is the following : The parity-odd contribution to the non-homogeneous part is again a contact term.

Renormalisation
In Sections 3 and 4 we saw that CFT correlators in spinor-helicity variables and momentum space are given by triple-K integrals of the kind : where K ν is a modified Bessel function of the second kind and α and β i are discrete parameters that depend on the dimension of space and the conformal dimensions of the operators. The integral is convergent except when the following equality is satisfied for any (or many) choice of signs [2,19,20,22] : When the integral is divergent, one regulates it by shifting the dimension of the space and the conformal dimensions [2,19,20,22] : where u and v i are four real parameters. This results in a shift in the discrete parameters of the triple-K integral indicated as follows : Here we note that when one deals with parity odd contributions to a correlator one has to set u = 0 as the Levi-Civita tensors are defined in the original dimensions and not in the shifted dimensions and hence one cannot use dimensional regularisation. For cases where the divergence condition (5.2) is satisfied for the choice of signs (−−−) or (− − +) or its permutations, one gets rid of the singularities in the regularised correlator by adding suitable counter-terms to the CFT action. For cases where the equality (5.2) is satisfied for the choice of signs (− + +) and its permutations or (+ + +) there are no appropriate counter-terms that one can add to the action. These correspond to cases where it is the triple-K representation of the correlator itself that is singular [19,20,22].
In the following we will show using the example of the JJO ∆ correlator that in the spinor-helicity variables the relation between the parity-even and the parity-odd parts continues to hold even after renormalisation. To renormalise the correlators, we first convert our answers in the spinor-helicity variables to momentum space expressions, cure the divergences and obtain finite answers in momentum space. We then convert the resulting correlators back to spinor-helicity variables.

JJO ∆
The momentum space expression for the even part of the correlator was given in (4.10). It can be checked that the discrete parameters in the triple-K integral satisfy the divergence condition (5.2) when ∆ ≥ 4. When ∆ = 4, the divergence condition (5.2) is satisfied for the choice of signs given by (− − −). A convenient scheme of regularisation is choosing v 3 = 0 and u = v 1 = v 2 = 0 [22]. The divergence terms are removed by adding the counter-term with three sources [22]: where µ is the renormalisation scale. After removing the divergence by choosing an appropriate a( ) such that the singular term in the regularised correlator is canceled, we get the following finite renormalised form factor : In spinor-helicity variables, the renormalised correlator takes the following form : Let us now discuss the parity odd part of JJO ∆ . The momentum space expression for the correlator takes the form in (4.14). For ∆ = 4, the two triple-K integrals are singular for the choice of signs (+ − −) and (− + −) respectively. To remove the singularity, we add the following parity-odd counter-term with two sources and one operator : After removing the divergences, the resulting form factor is given by : Note that the form factor B reno is related to the one in the even case A reno 1 (k 1 , k 2 , k 3 ) by the following simple relation : In spinor-helicity variables, the correlator again takes the same form as in the parity even case : Thus we have illustrated following the case of JJO 4 that the parity-even and the parityodd parts of the correlator are given by the same form factor even after renormalisation.

T T O ∆
Let us now consider the T T O ∆ correlator. The transverse and traceless part of the even part of the correlator is given by [22] : In d = 3, the solutions of the primary Ward identities were obtained to be [22] : One can easily check that for ∆ = 1, ∆ = 2, ∆ = 3 a subset of the triple-K integrals that appear in the form-factors above are divergent. A convenient regularisation scheme to work with is u = v 1 = v 2 and v 3 = u. For ∆ = 1, 2, 3 one does not have counter-terms to remove the divergences. It turns out that the divergences that appear in these cases are exactly cancelled by the primary constants determined by the secondary Ward identities [22].

T T O 4
T T O 4 deserves special discussion as this is the first case where a type-A anomaly could occur [22]. It was noticed in [22] that in the regularised correlator the pole in the regulator multiplies a degenerate combination of form factors in the numerator and hence the divergent form factors amount to a finite anomalous contribution to the correlator. Thus counter-terms are not essential to remove such divergences. It was also shown in [22] that the divergences in the regularised correlator and the anomaly can be removed using an appropriate counter-term with suitable coefficients. The form-factors of the renormalised correlator takes the following form (we present only the scheme independent part) : where We now convert this result in momentum space to the spinor-helicity variables and obtain : This precisely matches the result that we obtained for the correlator by directly solving the conformal Ward identities in spinor-helicity variables (3.18). For ∆ = 4 (or more generally ∆ ≤ 5) the triple-K integral in (3.18) is convergent and we get finite results for the correlator without any renormalisation. For ∆ ≥ 6 the above triple-K integral is singular and one needs to regularise and renormalise appropriately.

Weight-shifting operators
In this section we obtain correlators in momentum space using weight-shifting operators, following [12] and [50]. The operators that we will primarily use are given by and their permutations, where In [50] we obtained JJO ∆ odd using weight-shifting operators was discussed earlier . The seed correlator, in terms of triple-K integrals is given by : The correlator JJO ∆ odd is then obtained by the action of the parity-odd operator D 12 which raises the spin of the operators at points 1 and 2 and lowers the weight of the operator at point 2 : after using appropriate Schouten identities. It was seen that this reproduces the correct answer even for the case of ∆ = 4 and ∆ = 5 where JJO ∆ odd is divergent. As is explained below, it is better to start without renormalizing the seed correlator. The target correlator is renormalized at the end, in case it has divergences.

Subtleties associated with divergences
There are some subtleties to note when the correlators are divergent as observed in [50]. When the target and seed correlators both have the same kind of divergence, one can renormalize the seed correlator appropriately and use weight-shifting operators to get the correct result for the target correlator. When the target correlator is not divergent but the seed correlator is, this method does not always work. For example, it does not work when the seed correlator has a non-local divergence while the target correlator is not divergent. To illustrate this, consider the following sequence that also reproduces JJO 3 odd : The seed correlator has a non-local divergence and upon appropriate renormalization it is given by : If one calculates the r.h.s. of (6.7) using (6.8), it can be easily checked that the answer does not match the known result for JJO 3 odd . In fact, the final correlator obtained this away goes to zero upon using Schouten identities. The correct way to go about it is to put in the full, unrenormalized seed correlator into (6.7), which is given by This way, in the final answer, we can see that the divergences cancel upon using Schouten identities and we get the correct answer for JJO 3 odd .

T T O ∆ odd
In [50] we computed T T O 1 odd and T T O 2 odd using weight shifting operators. Here, we use these operators to calculate the correlator for any scaling dimension of the scalar operator. Also, our earlier analysis used a definition of the 3-point function that resulted in the correlator having a non-trivial Ward-Takahashi identity. Here, following [22] we redefine the 3-point function so that the correlator is completely transverse as this simplifies the calculations. The following sequence of operators reproduces T T O ∆ odd where the seed correlator is given by : This can be easily checked to give the correct result for the cases of ∆ = 1, 2, 3, 4 by calculating the correlator and converting it to its form in spinor-helicity variables. For ∆ = 3 + 2n, the seed correlator has a non-local divergence and one must also be careful about the subtlety mentioned in Section 6.1.1. The explicit momentum space expression for T T O 3 odd is given by where we have used Schouten identities and the degeneracy in (2.16) to ensure that the divergences cancel and to simplify the expression. The form factor is given 11 by After doing a change of basis, this can be shown to match the answer obtained from spinorhelicity variables and bulk gravity calculations.
In general, we have

T JJ odd
As we saw in (G.8) and (G.9), T JJ odd is completely transverse with respect to all the momenta. We start with O 1 O 2 O 2 as the seed correlator, which is given by where µ is the renormalization scale. 11 Like for JJO in (4.18) -this form factor suggests an apparent collinear divergence. However, one can easily check that the numerator of the correlator also vanishes in the collinear limit leaving the correlator finite. Hence the only pole is at E → 0 as expected.
The following sequence of operators reproduces T JJ odd is a spin-s projector. The explicit momentum space expression for the correlator is given in Appendix F.
Although this expression looks very different from the expression obtained earlier in (4.51), it can be shown that they are the same upon using Schouten identities. This fact can easily be seen by converting both the momentum space expressions to spinor-helicity variables where they exactly match.

T T T odd
We know from Section 4.2.3 that the non-homogeneous part of T T T odd is a contact term and can therefore be ignored. In momentum space, this means that we only need to calculate the transverse part of the correlator. This is given by : where we have added cyclic permutations to make the correlator manifestly symmetric. The explicit answer in momentum space obtained this way has many form-factors and is not very illuminating. However, one can use Schouten identities and the degeneracies in (2.16) and (2.17) to reduce the answer to just four form-factors, out of which two are independent.
This matches the answer obtained using spinor-helicity variables in (3.32).

Homogeneous part of general 3-point function using weight-shifting operators
We can use weight-shifting operators H ij and D ij to compute the homogeneous part of more general correlators. For example, consider J s J s J s where s is even. The parity-even and parity-odd parts of the correlator are given by : The seed correlator O 2 O 2 O 2 is given by where µ is the renormalization scale. Correlators involving scalar operator can also be written down as When s is odd in (6.18), we require additional weight-shifting operators apart from H ij and D ij . This is analogous to the way we write correlators in the next section using a finite number of conformal invariant structures.
Computation of the non-homogeneous part requires us to take linear combinations of different sequences of weight-shifting operators such that the Ward-Takahashi identity is saturated. See, for example, the computation of T T T even in [12]. Compared to the homogeneous part, this is much harder to do for a general correlator as we do not know the Ward-Takahashi identity for a general spin-s current.

CFT correlators in terms of momentum space invariants
The aim of this section is to write down CFT correlators derived in previous sections in terms of a few conformal invariant momentum space structures. Let us define These can be used as building blocks for writing down momentum space 3-point conserved correlators since they arise naturally in the expressions for such correlators 12 . There are some interesting relations among the above defined quantities. For example up-to degeneracies.

Homogeneous contribution
From the summary in E.1, we may now write the momentum space three-point functions in a compact manner using the above invariants. Let us note that we are concerned only with correlators which satisfy triangle inequality. To do this we divide the correlator into two different classes.
For this class of correlators we only require Q ij and S ij . Consider J s 1 J s 2 J s 3 such that s 1 ≥ s 2 ≥ s 3 , s 1 ≤ s 2 + s 3 . Then, we have Correlators involving scalar operators can also be written this way, where b ij = k i k j and c 123 = k 1 k 2 k 3 . The 3-point function involving ∆ = 1 is obtained by shadow transforming (7.7). One can also write the correlator involving a generic scalar operator dimension δ but we do not reproduce this here.
We require P 123 and R 123 as well when the sum of the spins is odd. For example, when s is odd, we have When s is odd, these are the only structures using which J s J s J s can be written. One can use (7.5) to substitute even powers of P 123 in terms of Q ij 's. Other correlators with s 1 + s 2 + s 3 = odd can also be considered similarily.

Non-Homogeneous contribution
We have discussed homogeneous contribution so far. The story for the non-homogeneous contribution is more complicated. We do not have a generic form of the WT identity to evaluate three point functions involving operators of arbitrary spin. For example, if we consider the solutions for J s O 2 O 2 as given in (4.8), there is no discernible underlying structure to these expressions. The numerator becomes increasingly complicated as we consider higher values of s and one cannot write these as the power of some simple structure. We can similarly identify some structures based on the answers for JJJ , JJT and T T T however those relations are not illuminating as in the case of the homogeneous part, (see (7.7)) and we do not present them here.

Some interesting observations
In this section we collect a few interesting observations about the correlators discussed so far. For the purposes of this discussion, it will be useful to write the correlators as in (2.14).

Contact terms
To properly understand correlators in momentum space it is very important to understand the contact terms which arises in both parity-odd and parity-even cases. For example JJJ correlation function has a contact term which is parity odd and is given by (4.40). Fourier transforming this to position space will give us a term of the form Another example of a correlation function where both parity even and parity odd part has contact term, let us consider T T T . The parity even contact term is given by [26] T T T even ∝ c T k 3 which when converted to position space gives contact term of the form Parity odd contact term is given 13 by (G.19) which becomes where z are polarization tensors. Once again we have not mentioned exact form of the contact term. Interestingly, for both parity even and parity odd part the contact term arises in the non-homogeneous contribution. One way to understand parity odd case is to look at (2.12). The right hand side of this equation for parity odd case is always a contact term for the correlator we have considered. For example, the transverse Ward identity for JJJ takes the form (G.2) which is a contact term. One can check the same explicitly for T T T as well as other correlators computed in previous sections. We expect on general grounds that J s 1 J s 2 J s 3 odd,nh is a contact term. To conclude, we observe that A. Contribution to the contact term comes from non-homogeneous part of both parity-even and parity-odd correlator. For parity-even it was observed in T T T only.
B. Parity-odd non-homogeneous piece of the CFT correlator is always a contact term.
However parity-even non-homogeneous piece can be nontrivial as is discussed in previous sections. The reader is referred to appendix G for a detailed discussion regarding whether the contact parity-odd terms can be set to zero by field redefinitions.

Relation between parity-even and parity-odd solutions
If we look at the correlator in momentum space, see section 4, there seem to exist no clear relationship between parity odd and parity even part of the correlator. However, as is seen section 2 and 3, there exist a remarkable relationship between them in spinor-helicity variables, namely up-to some signs and factors of i. Let us explain this in terms of some concrete equations.
To start, let us consider the anstaz where F 1 , G 1 and F 2 , G 2 are form-factors for the parity-even and parity-odd parts of the correlator. Both F 1 and F 2 satisfy the same non-homogeneous equation, see for example (C.28), (C.37). However, the form factors G 1 and G 2 satisfy a different non-homogeneous equation, see for example (C.29), (C.38) in appendix C. This difference is coming due to the different contribution of WT identity to parity-even and parity-odd parts for − − + helicity component 14 . This implies non-homogeneous contribution to parity-even and odd cases generally differ, whereas the homogeneous solution is always the same. This relation becomes even more nontrivial in the cases where there is a divergence in the correlator. For example, for JJO 4 the solution of the conformal Ward identity is given by However, the triple-K integral is divergent and one needs to regularise and renormalise the correlator. To do so we go to momentum space (see section 5). The renormalization procedure for even and odd parts is also completely different and we required quite different kinds of counter-terms. However, converting back the renormalized results in spinor-helicity variables, we remarkably obtained the same result again. This happens to all other correlators having divergences and it would be interesting to have a better understanding of this observation.

Manifest locality test
In the context of the cosmological bootstrap, a manifestly local test (MLT) was derived for wavefunctions of scalars of dimension 3 and gravitons in any manifestly local, unitary theory [66]. MLT imposes the following condition on such wavefunctions [66] : lim kc→0 ∂ ∂k c ψ n (k 1 , . . . , k c , . . . , k n ) = 0 (8.10) 14 For − − − helicity the WT identity contributes the same for parity even and odd case.
In the following we discuss how this analysis can be used for calculating CFT correlator Based on the symmetries of the correlator we write down the following ansatz for the correlator : where we take the following ansatz for the form factor : where the pole in k 1 + k 2 + k 3 = 0 can be argued on general grounds and the power of the pole is fixed by dilatation Ward identity. We will now fix the coefficients that appear in the ansatz by imposing manifest locality. With respect to one of the scalar operators we have : This gives the following relation between the coefficients : We can easily check that with these conditions, MLT with respect to the second scalar operator is also satisfied, i.e. : Let us now impose manifest locality with respect to the stress-tensor operator : This gives the following constraint c 5 = −c 1 . We now substitute the coefficients back into the ansatz to get : Notice that form factor in (8.17) matches explicitly with form factor presented in (4.9). We hope to come back to this in future for a better understanding of other 3-point functions.

A comparison between position and momentum space invariants
It is interesting to compare momentum space invariants discussed in section 7 and position space invariants introduced in [61,67]. To illustrate this, let us consider JJT even part. This is given by We refer the reader to [61] for details about the notation. We see that there are 4 structures. Demanding conservation equation for currents, we get two relation a 2 = −4a 1 , a 3 = − 5 2 a 1 which leaves two independent structures ) Furthermore, using WT identity we get a relation between a 4 , a 1 and the two-point function coefficient c j . Eliminating a 4 we obtain where c j appears in two point function of J µ . Let us emphasize that, (8.18) is built out of conformal invariant structures whereas (8.20) is built out of conformally invariant conserved structures 15 . In (8.20), we can identify the term proportional to a 1 as homogeneous and the term proportional to c j as the non-homogeneous contribution. Let us note that, for a generic correlator involving arbitrary spin-s currents, in general it is quite complicated to arrive at the analogue of (8.20) starting from more the readily obtainable expression (8.18). Moreover, finding the non-homogeneous term in position space is equally complicated. However, in momentum space we naturally obtain the analogue of (8.20) directly. In other words, in momentum space we naturally divide the answer into homogeneous and non-homogeneous contributions and the conformal invariant conserved structure is naturally built in.

Summary and Future directions
To summarize, we have systematically solved for 3-point CFT correlators involving higher spin conserved currents and scalar operators in three dimensions. Spinor-helicity formalism simplifies considerably the CWI based analysis of correlators. It solves the problems associated with degeneracy which makes direct computation in momentum space difficult. In these variables, we found that the homogeneous part of the correlator gets an identical contribution from parity-even and parity-odd parts. We were also able to write down momentum space correlators in terms of conserved conformally invariant structures. For some correlators (for example T T O 4 ) which are divergent in momentum space, a careful renormalization analysis is required. However, in spinor-helicity variables, we observed that it turns out we directly get the finite part of the correlator which does not require any renormalization. We also verified some of the results using weight-shifting operators. Below we discuss some future directions. 15 It is quite difficult to build conformally invariant conserved structures directly without first writing conformal invariants and then demanding conservation. Free theory generating functions defined in [61] might be of help, however it will be difficult to separate out the homogeneous and non-homogeneous contributions. However, in momentum space we directly get the conformal invariant conserved structures and getting conformal invariant structures without the WT identity constraint would be a more challenging task.
In this paper we focused exclusively on 3-point correlators of scalars and conserved currents with spin. If one considers spinning operators which are not conserved (no WT identity) then the approach has to be adjusted accordingly. Some preliminary momentum space results in this direction were obtained in [29]. This analysis is important as it is a useful first step for constructing general 4-point spinning conformal blocks in momentum space. The spinor-helicity formalism, extended as in [68] to account for scattering of massive particles in 4d, should be useful for this purpose.
It would be interesting to utilise the spinor-helicity formalism in the analysis of 4-point functions. The momentum space CWI approach for 4-point CFT correlators has been used in [30,35,44,53]. We have seen that the spinor-helicity expressions for 3-point correlators are much simpler compared to the momentum space ones and it is natural to expect a similar simplicity in the analysis of higher point correlators.
For correlators involving higher spin conserved currents the non-homogeneous part of the correlator requires the knowledge of Ward identity. It would be nice to find out if a general structure exists for the Ward identity which would then help us in getting the non-homogeneous part of correlation functions comprising operators with arbitrary spin. It would also be interesting to generalize our results to cases which break the triangle inequality and to cases where the current operators are not exactly conserved [64].
We observed in this paper the equivalence of parity-even and parity-odd parts of the 3-point correlator when expressed in spinor-helicity variables, and how this continues to hold in examples where, due to divergences, regularising and renormalising the correlator is required. It would be interesting to understand on general grounds why this is the case even though the counter-terms required in both cases are entirely different.
It would also be very interesting to understand the MLT condition [66] discussed in Section 8.3 starting from basic CFT principles. In Section 8.3, we used MLT condition to compute the non-homogeneous contribution to T OO correlator. It would be interesting to understand how to use MLT conditions to calculate the non-homogeneous contribution to a generic 3-point CFT correlator. It would also be interesting to understand how this condition can be used to constrain 4-point correlators.
Appendices A Spinor-helicity notation In this appendix we will quickly summarise the spinor-helicity variables for 3d CFTs. For more details see [3,26] . We first embed the Euclidean 3-momentum k into a null momentum vector k µ in 3+1 dimensions : such that k = | k|. Given the 4-momentum we express it in spinor notation as : where α andα are SL(2, C) transform under inequivalent (conjugate) representations of SL(2, C). However, in 3 dimensions one has an identification between the dotted and undotted indices. To see this let us consider the vector τ µ = (1, 0, 0, 0). In spinor-helicity variables : We can now convert dotted indices to undotted indices using the following tensor : We also introduce the barred spinors as follows : We then have the following relations between the 3-momentum and the spinors.
Since αβ is an SL(2, C) invariant, we can use it to define dot products between spinors.
It can be also be used to raise and lower indices on the spinors for which we will use the following convention.
The reader is referred to appendix B of [26] or appendix C in [12] which contains a set of useful relations between spinor brackets that will be used throughout the main text. Finally, we also define the following polarization vectors which when dotted with the momentum space expression of a correlator, gives its expression in spinor-helicity variables.

B Homogeneous & non-homogeneous vs transverse & longitudinal contributions
While computing momentum space correlation functions one often splits the correlator into its transverse and longitudinal parts [2]. In this paper we find it more useful to split correlators into their homogeneous and non-homogeneous parts as defined in sec. 2.3.1.
In this appendix we emphasise and illustrate through examples that the transverse and homogeneous parts of a correlator are not identical, and also that the longitudinal and nonhomogeneous parts are not identical. In particular, we will show that while the homogeneous part of a correlator is always transverse, the non-homogeneous part in general contains both transverse and longitudinal contributions and is proportional to 2-point function coefficients.
As an example consider T OO . The correlator is given by [2] T OO = T OO transverse + T OO longitudinal (B.1) where the transverse part is given by For example when the scalar operator O has scaling dimension ∆ = 1 the form factor is given by [2] The form-factor is proportional to the coefficient of the scalar two-point function The longitudinal part of the correlator for ∆ = 1 is We see that the full correlator is proportional to the two-point function coefficient c O . Thus in our terminology the full answer is non-homogeneous and there is no homogeneous contribution to it. To summarize we have Let us now consider the case of T T T . The full answer in the terminology of [2] is given by which can as well be split into homogeneous and non-homogeneous pieces as follows where we made the following identification We now give explicit identification of the homogeneous and non-homogeneous contribution.
To simplify the discussion, we make use of transverse, null polarization vectors that are contracted with the free indices of the correlator. The longitudinal term drops out and what remains are the transverse pieces. For convenience we reproduce it here [26,51] T T T even = C 1 c 123 where C T T is defined by the two-point function In the transverse correlator (B.10), the term proportional to C T T is non-homogeneous and the rest of it (the term proportional to C 1 ) is homogeneous. To summarize, the term that is dependent on the two-point function coefficient (fixed by secondary conformal Ward identity in the language of [2]) is the non-homogeneous contribution. From the dS 4 perspective the interpretation is that the term getting contribution from W 3 (term proportional to C 1 ) is homogeneous and the term getting contribution from Einstein-gravity √ gR (term proportional to C T T ) is non-homogeneous.
To conclude, the non-homogeneous part of the correlator can contain both transverse as well as longitudinal parts. From the dS 4 perspective as well, the origins of the homogeneous and non-homogeneous contributions are distinct.

C Details of solutions of CWIs for various correlators
In this appendix we provide details of the calculations related to solving conformal Ward identities (CWIs) in spinor-helicity variables.
The details of the conformal Ward identities for this case were already given in Section 3.1. Here we consider a few examples. The s = 1 and the s = 2 cases have already been computed in [2].
Contracting (C.6) with k 1 z −κ 1 and with k 2 z −κ 2 we get the following equations for the parity even part of the correlator (3.16) 16 : From the form of the ansatz for the correlator in (3.16) and since the conformal Ward identity takes the form in (C.6), the equations satisfied by the odd parts F 2 and G 2 of the correlator (3.16) are identical to those for the even parts F 1 and G 1 respectively. We note that the equation for F 1 (and F 2 ) is independent of the spin s. The dependence on the spin comes through the dilation Ward identity and is given by : The same equation is satisfied by F 2 as well. The equations (C.8) for G 1 (and G 2 ) do not have a non-trivial solution. Solving (C.7) and (C.9) we obtain the result in (3.17).

Examples
In

D Identities involving Triple-K integrals
In this section we obtain non-trivial identities involving triple-K integrals by matching our results obtained for the correlator in spinor-helicity variables to the results obtained for the same in momentum space after converting to spinor-helicity variables. Let us first consider the correlator JJO ∆ . We will work in a convenient regularisation scheme in which we set u = v 1 = v 2 = 0 and v 3 = 0. The momentum space expression for the correlator after converting to spinor-helicity variables takes the following form :

G Parity-odd contact terms and field redefinitions
Here we address the following question: is it possible to set the non-homogeneous part of the parity-odd contribution, which is argued to be a contact term always, to zero using some field re-definition or a suitable definition of the correlation function?
We will argue below that it is not always possible to set the non-homogeneous parity-odd contribution to zero. It should be emphasised that the parity-odd contribution to the nonhomogeneous piece is also proportional to the parity-odd two-point function coefficient.
Our main argument below is that the parity-odd contribution to the Ward identity is non-zero in general and hence this will contribute nontrivially to the non-homogeneous parity-odd three-point function. To augment our CFT argument, we also perform some simple dS 4 computations and identify interaction terms in the lagrangian which give rise to non-homogeneous parity-odd contributions. This serves as an additional check that the parity-odd non-homogeneous contribution cannot be set to zero. To illustrate these points let us consider a few examples.

JJJ
For the case of JJJ , irrespective of whether the functional derivatives involved in the definition of the correlator act on the measure factors or not, the correlator takes the same form [20]. Following [20] we have JJJ here = JJJ there (G.1) where "here" refers to the correlator as per the definition in [20] (where the functional derivatives do not act on the measure factors) and "there" refers to the correlator as defined in [2] (where the functional derivatives do act on the measure factors). The transverse Ward identity for the correlator is [2] k 1µ J µa (k 1 )J νb (k 2 )J ρc (k 2 ) = f adc J ρd (k 2 )J νb (−k 2 ) − f abd J νd (k 3 )J ρc (−k 3 ) This leads to the following parity-odd non-homogeneous contribution to the correlator J(z 1 , k 1 )J(z 2 , k 2 )J(z 3 , k 3 ) odd,nh = c JJ This is a contact term and can be explained from the dS 4 perspective using the following interaction term The three-point tree-level amplitude due to this interaction is given by Following the prescription of [7] the correlator is given by where the prefactor 1 E can be understood to come from the associated conformal time integral Thus we see that the non-homogeneous part of the correlator (G.3) is reproduced by the interaction F F in dS 4 . Such a contact term cannot be erased by any field re-definition. The dS 4 interaction (G.4) also contributes to the two-point function JJ as z 1 z 2 k 1 which is consistent with the contribution of F F to the non-homogeneous part of the three-point function.
There is no interaction term that one can have from the gravity side that contributes to the non-homogeneous parity odd part of T JJ . The only term that could possibly have contributed to this correlator is (G.4). However since this term is independent of the metric the contribution from it to T JJ is zero, which is consistent with the CFT computation in (G.10).
(G. 19) where We again see that the non-homogeneous contribution is proportional to the parity-odd twopoint function coefficient c T . From the dS 4 perspective this contribution to the correlator can be understood to arise from the W W interaction. The W W interaction also reproduces the parity-odd two-point function of the stress tensor (G.15).