Momentum space parity-odd CFT 3-point functions

We study the parity-odd sector of 3-point functions comprising of scalar operators and conserved currents in conformal field theories in momentum space. We use momentum space conformal Ward identities as well as spin-raising and weight-shifting operators to fix the form of these correlators. We discuss in detail the regularisation of divergences and their renormalisation using specific counter-terms.


Introduction
Conformal Field Theories (CFTs) have wide applicability in diverse areas of physics, and are central to our understanding of quantum field theory in terms of RG flows. While CFTs are well studied in position space and Mellin space, they are relatively less studied in momentum space. Recent works on aspects of momentum space CFTs include . CFTs in momentum space find applications in cosmology [38][39][40][41][42][43][44][45][46][47], condensed matter physics [48,49], study of anomalies [50][51][52][53][54], Hamiltonian truncation methods for strongly coupled field theories [55,56] and, of course, the conformal bootstrap program [17,21,26,57]. From the perspective of perturbative field theory which is naturally formulated in momentum space, it is of interest to study CFTs in the same setting. Flat space scattering amplitudes are, via AdS/CFT, directly related to the flat space limit of CFT correlators in momentum space [58] 1 . Studying momentum space CFT correlators can therefore shed light on the structure of flat space amplitudes. Interestingly, evidence for the double copy structure -which exists for flat space amplitudes -was seen directly in momentum space CFT 3-point correlators in [16,27]. An important simplification in momentum space is that 4-point conformal blocks can be constructed from 3-point functions in a relatively straightforward manner in momentum space [26,36]. Another significant application is in the cosmological setting where the CMB bispectrum which is a measure of non-gaussianity is given by the 3-point function in momentum space.
Three-point functions of scalar and spinning operators have a simple, well-known form in position space -this is most easily seen by going to the embedding space. However, their momentum space analogues are quite complicated. For example, the scalar 3-point correlator in momentum space is solved in terms of triple-K integrals which can involve divergences [2]. A careful treatment would require the regularisation and renormalisation of these divergences [2,4] 2 . The story of spinning correlators is even more complicated [2,3,9,12].
Parity-odd correlation functions in momentum space have received very little attention. In three-dimensions they appear naturally. Consider, for example, the free fermion theory in three-dimensions. The scalar primary operator with the lowest dimension is given byψψ, and it is odd under parity. A correlator with an odd number of insertions of this operator will be parity-odd. Another place where parity-odd structures arise naturally is CFTs with a broken parity. A prime example of such CFTs is Chern-Simons matter theories [64][65][66][67][68][69][70][71]. In [66], it was argued that in such theories : where J s is the spin s conserved current, the subscript odd indicates a parity-odd contribution to the correlator and α, β, γ are theory dependent constants. In three-dimensions 3 we cannot obtain the parity-odd part of the correlator from the free theory and we need to use CFT techniques.
There are other instances where parity-odd contribution might be interesting. For example, although parity-odd correlators are as yet unobserved, the CMB bispectrum -a measure of non-gaussianity -could contain parity-odd contributions. Such a contribution can arise, for example, from an inflationary action with higher derivative corrections such as cubic Weyl tensor terms 4 . Such terms could source contributions to the primordial graviton bispectrum, as first discussed in [38] using the spinor-helicity technique. In [73] it was shown that the parity-violating (odd) contribution in the non-gaussianity of the primordial gravitational waves CMB vanishes in an exact de-Sitter background, but exists in inflationary quasi de-Sitter where it is proportional to the slow-roll parameter. Other studies of parity-violating CMB bispectrum include [74][75][76]. Our results on the parityodd structures for various 3-point functions constrain the momentum space form of any parity-violating bispectrum that exists.
In this paper, we use two complementary approaches to determine the momentum space structure of parity-odd CFT 3-point functions. The first method is the more direct one and involves solving momentum-space Ward identities. This approach was developed and used for the parity-even sector in [2].
The second approach utilises spin-raising and weight-shifting operators. These have been used in the conformal bootstrap literature [77][78][79] and more recently in fixing the 2 In position space one could get rid of divergences by working at non-coincident points, but in momentum space one cannot do this and this leads to UV divergences. 3 In four dimensions, the 3-point function of stress-tensor is given by [72] T T T = a T T T FB + b T T T FF + c T T T Maxwell . 4 These are of the form W W 2 where W denotes the Weyl tensor and W its Hodge-dual.
form of cosmological correlators [46,47]. We will construct parity-odd spin-raising and weight-shifting operators in momentum space and use them on scalar seed correlators to generate parity-odd spinning correlators. The rest of this paper is organised as follows. In Section 2, besides setting up the notation and terminology, we outline the two different techniques that we use in this paper to determine parity-odd 3-point functions. We also briefly discuss the divergences that arise, and their regularisation. In Section 3, we give an overview of the possible parity-odd 3-point structures in embedding space in various dimensions. In Section 4, we use momentum space conformal Ward identities to fix the parity-odd part of JJO . In Section 5, we construct parity-odd spin-raising and weight-shifting operators in momentum space and use them to determine spinning correlators by their action on simple scalar seed correlators. We show that the results for JJO obtained using spin-raising and weight-shifting operators match the results for the same obtained in Section 4. In Section 6, we compute the parity-odd 3-point function of the spin-one conserved current using weight-shifting and spin-raising operators. In Section 7, we compute the form of the odd part of T T O ∆ for chosen conformal dimensions of O. In the free fermion theory, where ∆ = 2, we match the results obtained using spin-raising and weight-shifting operators with the answer obtained by an explicit computation in the free theory. In Section 8, we construct parity-odd spin-raising and weight-shifting operators in four-dimensions and use them to construct the parity-odd part of the non-trivial correlator JJJ in four dimensions. We conclude with a discussion in Section 9. In the appendices we elaborate on various technical details. In Appendix A, we review essential details on the embedding space formalism. In Appendix B, we discuss the basics of momentum space two-point functions. In Appendix C, we present the Schouten identities relevant to us. In Appendix D, we give some computational details of the results presented in Section 7. Appendix E gives the form of various parity-even spin-raising and weight-shifting operators. In Appendix F, we argue on grounds of permutation symmetry that certain 3-point correlators with spinning operators in four-dimensions vanish.

Two approaches to determining momentum space correlators
Determining correlation functions is a significantly harder task in momentum space than in position space. For parity-odd correlators this gets even more tedious. We will now discuss two different approaches to determining momentum space correlators. We also discuss certain subtleties and limitations associated with the two approaches.

Using Conformal Ward identity : Strategy
In the first approach, following [1,2] and [4,9,10,12,14] where parity-even 3-point functions were determined, we start with an ansatz of the form m A m (k i )T m for the correlator. Here T m are all possible tensor structures that are allowed by symmetry and A m are form factors which are functions of the momenta magnitudes (k i ). The form factors are constrained by permutation symmetries (if any) of the correlator and by momentum space Ward identities. The latter lead to partial differential equations which can then be solved to determine the form factors, up to undetermined constants that depend on the specific theory. In Section 4, we use this method to fix the parity-odd part of certain 3-point functions in momentum space. An excellent mathematica package that we found useful in these computations is [80].
Let us now describe the momentum space Ward identities associated with dilatation symmetry and special conformal transformations.

Dilatation and Special Conformal Ward identities
We will denote the n-point Euclidean correlation function of primary operators O 1 , . . . , O n by O 1 (k 1 ) . . . O n (k n ) . We suppress the Lorentz indices of the operators for brevity. The correlator with the momentum conserving delta function stripped off is denoted as : (2.1) An n-point correlator with scalar or spinning operator insertions satisfies the following dilatation Ward identity [2] : This constrains the correlator to have the following scaling behaviour : The special conformal Ward identity on an n-point correlator with both scalar and spinning operators is [2] : In the second line of the RHS of the above equation, the indices of the generator mix with the spin indices of the correlator. In principle, one can solve this equation and get the desired correlator [2]. However, for parity-odd structures in three-dimensions, the computation gets complicated and has not yet been done. We will always be working with correlation functions with the momentum conserving delta function stripped off. From here on we will drop the double angular brackets notation to avoid clutter and use single angular brackets everywhere.

Divergences
Triple-K integrals arise as solutions to primary conformal Ward identities which are second order differential equations [2]. Along with the three momenta, they are expressed in terms of four other parameters : where K β j is a modified Bessel function of the second kind. While the integral is well behaved at its upper limit, it is convergent at x = 0 only if [2,9] : When the integral is divergent one can regulate it using two parameters u and v [2,9] : The regularised triple-K integral is convergent except when [2,9] : for any choice of signs. When (2.8) is satisfied, the integral is singular in the regulator and we will denote the divergence by the choice of signs (± ± ±) for which (2.8) is satisfied. Divergences of the type (− − −) are called ultra-local and they occur when all the three operators are co-incident in position space. In momentum space, this manifests as the divergent term being analytic in all three momenta squared. Such divergences must, in general, be removed using counter-terms that are cubic in the sources, and they give rise to conformal anomalies.
Divergences of the type (− − +) and its permutations are called semi-local divergences. In position space, this is a divergence that occurs when two of the operators in the correlator are at co-incident points. In momentum space, the divergence is said to be semi-local when the O(1/ ) term is analytic in any two of the three momenta squared. In general, these divergences must be removed by counter-terms that have two sources and an operator. Such terms lead to non-trivial beta functions.
Divergences of the kind (+ + +) and (+ + −) are non-local and they occur even when all three operators are at separated points in position space. In momentum space, such a divergence is analytic in at most one of the momenta squared. This is not a physical divergence and arises because the triple-K integral representation of the correlator is singular. In this case no counter-term exists and the divergence is removed by imposing the condition that the constant multiplying the triple-K integral vanishes as an appropriate power of .

Counter-terms
As we discussed above, divergences of the kind (− − +) and (− − −) that correspond to ultra-local and semi-local divergences are removed using suitable counter-terms. In the case of parity-even correlators this has been extensively studied in [2,4,9,12].
We will now list a few potential counter-terms that could turn out to be useful in our study of parity-odd correlators. For ultra-local divergences, for example we have : and for semi-local divergences : where F 3 (A) is the Chern-Simons form in three-dimensions given by, C µν is the Cotton-York tensor given by, and R µν and R are the Ricci tensor and the Ricci scalar respectively. In the above list of possible counter-terms (2.10) we have included certain parity-even terms such as d 3 x A µ J µ n φ and d 3 x F µν J µ D ν φ. These counter-terms could give rise to the 2-point function of currents, which has a parity-odd contribution J µ (p)J ν (−p) ∝ µνρ p ρ .

Using Weight-shifting and Spin-raising operators
The second method of computing correlation functions in momentum space hinges on the technique of weight-shifting and spin-raising operators. In position space, this technique was initiated in [78] and extensively developed in [79]. In this approach, starting from certain seed correlators, the action of conformally covariant weight-shifting and spin-raising operators generates the desired correlator.
To describe this method in some detail, let us consider a spinning correlator J s 1 J s 2 J s 3 . The first step is to count the number of independent tensor structures associated with this correlator. For parity-even correlators this number in position space is given by [77] : where p = max(0, l 1 + l 2 − l 3 ). The second step is to consider a seed correlator of the form O ∆ 1 O ∆ 2 J s 3 and find out N + 3d (l 1 , l 2 , l 3 ) ways to reach J s 1 J s 2 J s 3 . This involves acting upon the seed correlator with various spin-raising and weight-shifting operators.
In momentum space, we are constrained in our choice of seed correlators because correlators of the form J (l) O ∆ 1 O ∆ 2 , where J (l) is a spin-l conserved current, are non-zero only when ∆ 1 = ∆ 2 . A more convenient approach was recently advocated in [46,47] to compute (parity-even) spinning cosmological correlators where instead of starting from the seed , and apply spin-raising and weight-shifting operators such that the resulting correlator saturates the Ward-Takahashi identity. See Section 4.2.2 of [47] for an example.

Subtleties with the weight-shifting and spin-raising operator approach
In momentum space, one must consider the types of divergences in the seed and target correlators. It is not always possible to reach a target correlator starting from a seed correlator although a naive application of the spin-raising and weight-shifting operators might suggest so. This is most easily understood in the case of scalar correlators. As a concrete example of such a situation, consider the following two correlators in threedimensions : where µ is the renormalisation scale. Although it might seem like we can use the weightshifting operator W ++ 12 (defined in (E.4)) to go from the first correlator to the second, this is clearly not possible as The above example tells us that weight-shifting operators fail to reproduce the correct correlators when the divergence type changes from non-local to semi-local or ultra-local. The conditions for various types of divergences, in terms of scaling dimensions of the operator insertions, are given by : where k 1 , k 2 , k 3 , k 4 ≥ 0. We can see that the only time the divergence structure changes is when k i = 0. For the non-local cases in three-dimensions, these correspond to the following for the seed correlator : When either of these conditions is satisfied by the seed correlator, the action of W ++ 12 does not reproduce the correct result. However, W −− 12 works as it can be checked that it does not change the type of divergence.

Allowed parity-odd 3-point functions in various dimensions
Parity-odd structures for three-point correlators can exist in simple theories such as the free fermion theory in three-dimensions and in CFTs which do not have a parity symmetry.
Such structures change sign under inversion and they always involve the antisymmetric epsilon tensor. The existence, or non-existence, of such correlators in various dimensions is easily seen in the embedding space formalism [77,78]. In embedding space, d-dimensional odd-correlators are characterised by (d + 2)-dimensional epsilon tensor. Although we restrict our attention to correlators involving spin-zero, spin-one and spin-two currents, the analysis below can be easily generalised to parity-odd correlators involving traceless symmetric operators in various dimensions. We refer the reader to Appendix A for some details on the embedding space formalism.
One of the constraints on correlators with a spinning operator is that they are transverse, i.e. the epsilon structure should be invariant under Z i → Z i + βX i . This constrains the possible epsilon structures one can have in a given dimension.

Three-dimensions
In three-dimensions, transversality implies that the parity-odd invariants that can exist are: This immediately implies that the following parity-odd correlation function is zero : where O ∆ 1 , O ∆ 2 are scalar operators with dimensions as indicated and J s is a spin s operator. However, correlators of the form J s 1 J s 2 O ∆ odd and J s 1 J s 2 J s 3 odd are non-zero. See [81] for details. In Section 4.1, we will explicitly show that, in momentum space : We will also calculate other parity-odd three-point functions in subsequent sections.

Four-dimensions
In four-dimensions, transversality allows only the following parity-odd invariant : This implies that in four-dimensions the following correlators are zero :

Five-dimensions and above
In five-dimensions, the only parity-odd invariant allowed by transversality is : The structure of the contracted epsilon tensor makes it clear that we cannot have any odd three-point function in five-dimensions 5 . At this point it is also clear that we cannot have an odd three-point function in d ≥ 5, or an odd four-point function in d ≥ 6.

Momentum space parity-odd 3-point functions in three-dimensions using conformal Ward identity
In this section, we will use the direct approach of solving the PDEs for the form factors which result from the momentum space conformal Ward identities. We will illustrate this using the correlators JOO and JJO as examples. This is an extension of the analysis in [2] to the parity-odd sector. We also discuss the divergences that can arise, and the counter-terms that regulate them. We conclude this section with a small discussion on the difficulties in using this method to compute correlators involving more general spinning operators.

J µ OO odd
Let us consider the parity-odd part of the J µ OO correlator where J µ is a conserved current and O has scaling dimension 2. A suitable ansatz for the correlator is : Throughout this paper, we use notations such as µνk 1 and µk 1 k 2 and they stand for the epsilon tensor contracted with the momenta : We have considered J µ to be in the third position as this makes the action of K κ simpler (2.4). Without non-abelian indices the correlator is zero as (k 1 ↔ k 2 ) exchange symmetry would require A(k 1 , k 2 ) = −A(k 2 , k 1 ). After acting with K κ , the primary Ward identities are given by : 5 However, the above odd structure hints at the possibility of odd four-point correlators in five-dimensions.
Since there exists no parity-odd three-point function in five-dimensions, the parity-odd four point function will be a contact term.
The above differential equations can be solved in terms of triple-K integrals (2.5) to get : The correlator also satisfies an independent secondary Ward identity given by : The right hand side of the above equation is proportional to the 2-point function However, the scalar two-point function has no parity-odd contribution and thus the R.H.S. of (4.5) is zero. Substituting (4.4) into (4.5) gives Thus we conclude that : This result can be generalised to scalar operators of arbitrary scaling dimensions.
Here we will consider the parity-odd part of the correlator J µ J ν O . We start with the following ansatz for the correlator : where the orthogonal projector π µ ν (p) is given by : The ansatz (4.8) is chosen such that the correlator is transverse with respect to k µ 1 and k ν 2 . Demanding symmetry under the exchange : (k 1 , µ) ↔ (k 2 , ν) gives the following relation between the form factors : Using the definition of projectors (4.9), the ansatz (4.8) expands to the following : where we have used momentum conservation to choose k 1 and k 3 as the independent momenta.
We now use Schouten identities (C.3) and (C.4) to get rid of the µk 1 k 2 tensor structure and re-express the ansatz in (4.11) as : where the new form factors A(k 1 , k 2 , k 3 ) and B(k 1 , k 2 , k 3 ) are given in terms of A(k 1 , k 2 , k 3 ) and B(k 1 , k 2 , k 3 ) as follows : Note that the exchange symmetry (4.10) continues to hold between A and B: We will now obtain the primary and secondary Ward identities that A(k 1 , k 2 , k 3 ) and B(k 1 , k 2 , k 3 ) satisfy, by letting the generator of special conformal transformations K κ (2.4) act on the ansatz (4.12) : Note that by choosing k 1 and k 3 as the independent momenta, we got rid of one set of terms in the generator K κ that mixes with the index structure of the correlator. The primary Ward identities satisfied by A(k 1 , k 2 , k 3 ) are given by : Similarly, the equations for B(k 1 , k 2 , k 3 ) are given by : The general solution to both the primary Ward identities can be found in terms of triple-K integrals (2.5). We solve for β 1 , β 2 , β 3 by substituting the triple-K integral into the primary Ward identities, and obtain : The unknown α is determined using the dilatation Ward identity. The action of the dilatation Ward identity on the ansatz gives the degree of the form factors : where N A and N B are the tensorial dimensions of A and B, defined as the number of momenta that multiply the form factor in the ansatz. We see from (4.12) and (4.8) that Similarly, we impose the dilatation Ward identity on the triple-K integral and get : This must equal the degree of the form factors A and B (4.19) giving us : Thus we obtain : where c 1 and c 2 are undetermined constants. We now present the explicit expressions for the two form factors for a few values of the scaling dimension of the scalar operator O. When the scalar operator has ∆ 3 = 1, we have, When ∆ 3 = 3 : (4.25) We will now look at the secondary Ward identities to fix the undetermined constants c 1 and c 2 in (4.22).
There is one independent secondary Ward identity in this case which leaves just one independent, undetermined constant. The identity is given by : Substituting the solutions for the form factors from (4.22) in this equation we get: which is exactly what is expected from symmetry considerations.

Divergences and Renormalization
We saw in equations (4.23), (4.24) and (4.25) that the triple-K integral is convergent for ∆ = 1, 2, 3. For ∆ 3 > 3, the integral is singular in the regulator and in some cases, we will require counter-terms to remove this divergence. The generating functional for the theory is defined as : where φ and A µ are sources of the scalar operator and the conserved spin-one current respectively. For certain classes of divergences, the generating functional is modified by counter-terms. We classify the values of ∆ 3 into two classes based on the kinds of divergences that occur. ∆ 3 = 4 + 2n : When ∆ 3 = 4 + 2n where n ∈ {0, 1, 2, ..}, (2.8) is satisfied for the choice of signs given by (+ − −) for n = 0, i.e., ∆ 3 = 4. When n > 0, it is satisfied for the choice of signs (+ − −) and (− + −).
To remove this singularity, we look at the following parity-odd counter-term from (2.10) where µ is the renormalization scale. After taking suitable functional derivatives, the contribution to the correlator from this counter-term is given by A Fourier transform of the above gives : where we used the following 2-point function 6 : Using Schouten identities (C.3) and (C.4), the ansatz for the correlator can be written as When ∆ 3 = 4 the singular part of the regularised form factors are given by The contribution of the counter-term (4.31) to the correlator in this case (∆ 3 = 4, or equivalently n = 0) is given by : Comparing (4.35) and (4.33) along with (4.34) we see that choosing a( ) = 1/ cancels the singular part of the correlator. After removing the divergences, the resulting form factor is given by : The second form factor is obtained by the following exchange : The anomalous dilatation Ward identity takes the form : ∆ 3 = 5 + 2n : When ∆ 3 = 5 + 2n where n ∈ {0, 1, 2, ..}, (2.8) is satisfied for the choice of signs given by (− − −) and (+ + −). Although we have both an ultra-local and a non-local divergence here, the term at O(1/ ) is non-local in the momenta and therefore the divergence can be cancelled by multiplying with a constant of O( ) and then taking the limit → 0. In particular, when ∆ 3 = 5, the divergent term can be calculated to be : Choosing c 1 to be O( ), the resulting form factor is : where c 1 is O(0) in . It can be easily checked that this form factor satisfies non-anomalous Ward identities and that scale invariance is not broken.

Non-Abelian JJO odd
If we add a colour index to all the three operators in the correlator such that an exchange of two colour indices gives rise to a minus sign, then the symmetry between the form factors in (4.8) changes to : The solution for J µa (k 1 )J νb (k 2 )O c ∆ (k 3 ) , when ∆ = 3 is then given by : It can be checked that the above solution is symmetric under (1 ↔ 2) exchange upon using suitable Schouten identities. In principle one can compute parity-odd correlation functions of higher spin operators using the approach described in this section following [2]. However, it soon gets difficult to find out the independent tensor structures after the application of the generator of special conformal transformations, due to non-trivial Schouten identities that relate various tensor structures. We will now resort to the technique of using weight-shifting and spin-raising operators to compute parity-odd correlation functions.

Parity-odd spin-raising & weight-shifting operators in three-dimensions
In this section we construct parity-odd spin-raising and weight-shifting operators in momentum space. We then illustrate how these operators can be used to calculate the parity-odd part of the JJO correlator.

Parity-odd operators
We consider a parity-odd operator which raises the spins of the operators at points 1 and 2 and lowers the weight of the operator at point 2. In embedding space, this operator is defined as based on requirements of transversality and interiority [78]. In position space, the operator takes the form : where ijk−+ ≡ ijk . Performing a Fourier transform we get in momentum space : where K + 12 and K − 12 are defined in Appendix E. The above operator acts on a momentum space correlator with a momentum conserving delta function. In its present form, it will be tedious to take the above operator past the delta function. Consider now the following commutator : The above commutator vanishes on a three-point function due to momentum conservation. Thus we have the following action on 3-point functions : We contract the above equation with ∂ ∂k µ 1 from the right to get the Schouten identity : We use this to rewrite (5.3) as : where W −− 12 is defined in Appendix E. Note that the operator defined in (5.7) is explicitly translation invariant 7 .
We will use D 12 in (5.7) to compute JJO ∆ odd , JJJ odd and TTO ∆ odd 8 starting from a scalar-seed. We can also construct D 23 and D 31 to get operators that act on points 2 and 3 and points 3 and 1, respectively. We will require them in the computation of JJJ odd as the correlator has cyclic symmetry.
We will now construct other parity-odd weight-shifting and spin-raising operators that are useful. Let us consider the following : The Fourier transform of this operator gives in momentum space the following : The above can be written in a translation invariant manner using the methods implemented in the case of D 12 to obtain : We also introduce the following parity-odd operator : Translational invariance of an operator implies in momentum space that its operation on the momentum conserving delta-function is zero. This happens when the operator is only a function of K − ij in the derivatives, and that is precisely what we have in (5.7). 8 The subscript on operators denotes their bare scaling dimensions.
which after a Fourier transform takes the form : This operator is naturally translational invariant. Finally, the polarization vectors can be stripped off from the correlators using the Todorov operator [82] : We will now discuss the computation of JJO odd using the technique of spin-raising and weight-shifting operators.

Reproducing JJO odd correlators
We start with the seed correlator O 2 O 3 O ∆ , where the last operator has an arbitrary scaling dimension ∆. To get JJO ∆ odd we act on the scalar-seed with the parity-odd operator D 12 defined in (5.7) : Let us consider the case when ∆ = 2. We have : Acting with D 12 and then removing the polarization vectors gives This matches the answer in (4.12). To see this explicitly, we substitute the solution to the form factors given in (4.24) and (4.27) in the ansatz (4.12) : Using the Schouten identities in (C.3) and (C.4) to replace νk 1 k 2 in the above equation, we match the result obtained using weight-shifting and spin-raising operators in (5.16). Similarly, one can obtain and match the results for JJO 1 odd and JJO 3 odd . We were also able to obtain JJO ∆ for ∆ = 1, 2, 3 using the operator in (5.9). However, for ∆ = 3 we obtained the correlator up to a conformally invariant contact term given by (z 1 z 2 k 1 ).
When the dimension of the scalar operator is greater than or equal to 4, one needs to be more careful as the correlators are divergent and need to be renormalised (see Section (4.2.1)).

5.2.2
The full seed correlator along with the divergent part is given by : This correlator has two semi-local divergences labelled by (+ − −) and (− + −). The generating functional is given by : where φ 0 , φ 1 and φ −1 correspond to the sources of O 3 , O 2 and O 4 respectively. The counter-term in this case is given by The full renormalized correlator is then defined by We now act D 12 from (5.7) on the above renormalised scalar correlator and then strip off the polarization vectors to obtain : where A 1 is as in (4.36) and A 2 (k 1 , k 2 , k 3 ) = −A 1 (k 2 , k 1 , k 3 ). Once again, using the Schouten identities in (C.3) and (C.4), this matches the solution (4.33) and (4.36) obtained by solving the conformal Ward identities.

5.2.3
The seed correlator here has ultra-local and non-local divergences. The correlator is given by Unlike the previous case, this divergence is removed by taking the constant c 1 to be O( ) and then taking the limit → 0 (similar to (4.39) and (4.40)). Thus we have the renormalised correlator : The rest of the calculation is the same as in the case of JJO 4 odd and we get where A 1 is as in (4.40) and A 2 = −A 1 (k 1 ↔ k 2 ). It can be easily checked that this matches the answer obtained by solving conformal Ward identities (see (4.33) and (4.40)).

JJJ odd in three-dimensions
We now turn our attention to computing the odd part of the JJJ correlator. The correlator is non-zero only when the currents are non-Abelian. We express JJJ odd in terms of transverse and longitudinal parts [2] : where j µa j νb j ρc odd denotes the transverse part of the correlator. The ansatz for this part of the correlator can be written as Calculating the form factors by directly solving the conformal Ward identities is quite complicated. Here we will instead use spin-raising and weight shifting operators to calculate them.
Starting from the seed correlator we can get JJJ odd by : where D 12 is defined in (5.7) and D 11 in (E.5). The renormalized seed correlator is : This gives : We can now read off the form factors in (6.3) by comparing it with (6.6) : The JJJ odd correlator obeys the following Ward-Takahashi identity [2] : Our result does satisfy this identity. To see this, let us contract our result (6.1) for J µa J νb J ρc with k µ 1 : Using the identities in (C.6) and (C.7), we can get rid of those epsilon structures which have two momenta contracted with their indices. Doing so we obtain : which matches the desired Ward identity (6.8).

T T O odd in three-dimensions
We now turn our attention to the T T O odd correlator. Based on symmetry considerations and conservation, this correlator is expected to take the following form [2] : and t µ 1 ν 1 (k 1 )t µ 2 ν 2 (k 2 )O I (k 3 ) odd is the transverse part of the correlator. Due to symmetry and transversality, this is expected to take the following form : where the traceless-orthogonal projector is given by : In the following, we first present a direct calculation of T T O in the free-fermion theory for which O =ψψ which is parity-odd in three-dimensions and has scaling dimension ∆ = 2.
We then reproduce the free theory answer using spin and weight-shifting operators. We also compute T T O odd with ∆ O = 1 which does not have a free theory analogue 9 . We defer the computation of T T O ∆ for general ∆ to future work.

T T O 2 for free fermionic theory
The free-fermion action in curved space is given by [2] : where e a µ are the vielbeins and the covariant derivative acts as follows on the spinor One may use this action to compute the stress-energy tensor [2] : which after taking the flat-space limit and a Fourier transform gives the stress-energy tensor in momentum space : The parity-odd scalar primary in the free-fermion theory is given by : Using these definitions it is straightforward to evaluate T T O 2 for the free-fermionic theory.
Since O 2 is parity-odd T T O 2 is also parity-odd. We give the details of the computation in Appendix D and give only the final results here. The form factors that appear in the transverse part of the correlator (7.3) are : To compute the longitudinal part, we require the functional derivative of the fermionic stress-energy tensor. Using the action of the covariant derivative on spinors (7.6) we reexpress the stress-energy tensor (7.7) as : After some computation (see Appendix D.2) we obtain : Taking a Fourier transform of the above we get : Using the above, one may compute in the free-fermion theory. This can now be used in the reconstruction formula 7.1 to get the full correlator T T O 2 .
Having computed the functional derivative, we now give the trace and transverse Ward identities associated with T T O 2 . They are [2] : The expression obtained in (7.14) is traceless in (µ, ν) and (α, β). This immediately implies that the trace Ward identity is trivial : Contracting (7.14) with k µ 1 gives the transverse Ward identity : In Appendix (D.1) we show that precisely this Ward identity holds.

T T O 2 odd using parity-odd spin-raising and weight-shifting operators
In this section we compute the odd part of T T O 2 using spin-raising and weight-shifting operators. We start from the renormalised scalar-seed correlator given by : The following sequence of operations gives the T T O 2 correlator : where a i and b i are coefficients which are fixed by comparing with the result for the correlator from the explicit computation in the free fermion theory : It can be checked that the operators in (7.20) are not all independent. One has the following linear dependence between them : Using these relations and setting b 4 = 1 180 one obtains the odd part of T T O 2 correlator from the following :

7.3
T T O 1 odd using parity-odd spin-raising and weight-shifting operators In this subsection we discuss the parity-odd part of the correlator T T O 1 . To construct this correlator we start from the renormalised scalar seed correlator given by : Note that this is related to the scalar-seed O 4 (k 1 )O 5 (k 2 )O 1 (k 3 ) in (7.19) which was used to compute T T O 2 as : The correlator T T O 1 is given by the same sequence of operations (7.23) acting on Note that the spin-raising and weight-shifting operators that appear in the above equation are independent of the third momentum and hence do not act on the k 3 pre-factor in (7.25).
As a result we obtain the following relation 10 :

Momentum space parity-odd 3-point correlators in 4d
In this section we will construct and use parity-odd spin-raising operators in four-dimensions to determine the non-trivial momentum-space correlator JJJ . In four-dimensions, the only non-zero parity-odd 3-point function is JJJ with a nonabelian current J. Other correlators involving operators with spin s ≤ 2 are zero as emphasized in subsection 3.2.

Spin-raising and weight-shifting operators
In four-dimensions, transversality and interiority allow for only one operator : where the embedding space result has been converted to ordinary position space. Fourier transform of the above operator gives : Let us now consider the following commutator : which is zero by momentum conservation. The above commutator gives an operator-based Schouten identity. We find the following contracted form of the above equation useful in the present context : The above contracted form allows us to write a manifestly translation invariant form of the spin-raising operator : As in (5.7) the operator depends only on K − ij in the derivatives, ensuring translational invariance.

JJJ odd
In this section we use the operator in (8.5) to derive the odd part of JJJ . In fourdimensions, this correlator is expected to have the following structure based on transversality and momentum conservation : The above ansatz can be simplified using the Schouten identities in (C.8) to the following Due to cyclic symmetry we obtain the following relations between the form factors This shows that there is only one independent form factor. To compute JJJ odd in fourdimensions one has to compute : The correlator O 3 O 3 J has the following ansatz : where OOj is the transverse part, which has the following ansatz The form factor A is computed by solving the conformal ward identities, and is given by We substitute this in the reconstruction formula (8.10) since D ++ 12 kills the longitudinal part. We now compute (8.13) to get the following explicit form of the form factor :

Discussion
In this paper we computed momentum space parity-odd 3-point functions of scalar operators and conserved currents in a CFT. We used conformal Ward identities to fix the parity-odd part of the JJO correlator. While using this method to fix more complicated 3-point functions of spinning operators such as JJJ odd or T T T odd , we faced the difficulty of identifying independent tensor structures after the application of the generator of special conformal transformations on the ansatz for the correlator. The difficulty arises from Schouten identities that relate the various tensor structures. We leave this problem to a future work. We defined parity-odd spin-raising and weight-shifting operators, and used them to compute JJJ odd and T T O ∆ odd for ∆ = 1, 2. However, we found it quite complicated to generalise this analysis to obtain correlators such as JJT odd and T T T odd . The difficulty arises from the large number of possible paths to reach a specific correlator starting from a seed correlator. Another difficulty was the difference in the singularity structure of the seed correlator and the correlator of interest to us. We leave further investigation for the future.
Some of the directions that we would like to pursue in the near future are the following. It will be interesting to extend the analysis of [16,27] and study the double copy structure of parity-odd correlation functions [83]. It will be interesting to generalise the construction of momentum space conformal blocks in [36] to the case where parity-odd contributions are important. In momentum space, conformal blocks are constructed quite simply by taking products of 3-point functions of primary operators (contrasted against position space, where an infinite sum over conformal descendants is required) [23,26]. Recently, in the study of cosmological correlators [43,46,47], the form of tree-level four-point functions in momentum space was constrained. Tree level spinning correlators such as JJOO can get parity-odd contributions as they are built out of products of three-point functions. It would be interesting to find the explicit form of these and study their physical implications. Another interesting direction to pursue is to understand the parity-odd structure of 3-point correlators with operators of arbitrary spin. To do this, in addition to the techniques used in this paper, one could use the constraints imposed on the correlators by higher spin equations [32,35]. Solving higher spin equations requires us to include possible contact terms in the correlator. It would be interesting to classify both parity-even and parity-odd contact terms for a given spinning correlator. One can also study momentum space correlation functions of spinning operators in supersymmetric theories [84,85].

Acknowledgments
We thank Nilay Kundu and Vinay Malvimat for discussions and collaboration at an early stage. The work of SJ and RRJ is supported by the Ramanujan Fellowship. AM would like to acknowledge the support of CSIR-UGC (JRF) fellowship (09/936(0212)/2019-EMR-I). The work of AS is supported by the KVPY scholarship. We acknowledge our debt to the people of India for their steady support of research in basic sciences.

A Embedding space formalism
In this section we briefly review some aspects of the embedding space formalism following [77]. Conformal invariance is most manifest in the embedding space formalism as the d-dimensional Euclidean conformal algebra is the (d + 2)-dimensional Poincare algebra. The d-dimensional CFT correlator in position space can be written in terms of (d + 2)dimensional embedding space. The embeddding space coordinates X M are defined in terms of the position space coordinates x i as follows : where the Z M and z i are the null polarization vectors in the embedding space and position space, respectively. The derivative on the embedding space is defined as follows The (d + 2)-dimensional space metric is as follows The embedding space coordinates are defined such that Z · X = X 2 = Z 2 = 0 because the CFT is defined on the null light cone of the AdS space. The spin-raising and weight-shifting operators that we study in this paper are constructed to have two properties in the embedding space, namely transversality and interiority. Transversality requires that under the transformation Z i → Z i + βX i , the operators remain invariant, while under interiority, the operator must map null light cone to itself. The operators thus constructed are manifestly invariant under conformal transformations.
In the main text we have often used notations such as (Z 1 , Z 2 , X 1 , X 2 , X 3 ) : B Parity-odd two-point functions As is well known, scale invariance completely fixes CFT two-point functions. parity-odd structures can exist for two-point functions of spinning operators.

B.1 Four and Higher dimensions
In four or higher dimensions, it is not possible to have any parity-odd two-point function of either spin-one or any other spinning symmetric spinning correlator. This is because a parity-odd correlator must necessarily involve the tensor and it is simple to show that it is impossible to have any parity-odd 2-point function of a spin-1 or any symmetric tensor operator.

B.2 Three-dimensions
In three-dimensions parity-odd two-point functions exist. These come from purely contact terms 11 . We will look at the parity-odd 2-point functions of spin-one and spin two conserved currents.
The general ansatz for the correlator is given by The ansatz guarantees that the correlator is transverse to the momentum. Imposing scale invariance gives the following differential equation for the form factor A(k) : This implies that the form factor is just a constant in this case and we have : We will now consider the parity-odd 2-point function of the stress-tensor.
We consider the following ansatz for this correlator : where ∆ µνρσ (k) is a parity-odd, transverse-traceless projector given by : where π µ ν (k) is the same projector used in previous sections. The ansatz guarantees that the correlator is transverse and traceless. The dilatation Ward identity gives the following equation for the form factor B(k) : This can be easily solved to get Therefore, the correlator is given by

C Schouten Identities
Here, we list the Schouten identities used in our calculations in the main text. The most general form of a Schouten identity in d-dimensions is In three-dimensions, this translates to Dotting the indices in (C.2) with momenta lets us relate different epsilon structures that occur in the correlation functions calculated earlier. Dotting with k 1µ 3 , k 1ρ and k 2ν gives µ 1 µ 2 k 1 (k 1 · k 2 ) + µ 1 k 1 k 2 k µ 2 1 = µ 1 µ 2 k 2 k 2 1 + µ 2 k 1 k 2 k µ 1

(C.7)
In four-dimensions, we use the following identities to rewrite the ansatz for JJJ .

E Parity-even spin-raising and weight-shifting operators
In this section we list out all the parity-even weight-shifting operators used in the main text of the paper [46,47]. The operator that decreases the scaling dimension of operators at points 1 and 2 is : We also use We can also define an operator that increases the scaling dimension at 2-points. Although this has a very complicated expression, it simplifies when acting on scalar operators and is given by : D 11 raises the spin of the operator at point 1 and simultaneously lowers its weight. This was used in the construction of both T T O and JJJ : We can similarly define D 22 and D 33 by doing cyclic permutations of the momenta and polarization vectors in (E.5). For example, D 22 ((k 1 , z 1 ), (k 2 , z 2 ), (k 3 , z 3 )) = D 11 ((k 3 , z 3 ), (k 1 , z 1 ), (k 2 , z 2 )) (E.6) S ++ 12 raises the spin at points 1 and 2 : S ++ 12 =(s 1 + ∆ 1 − 1)(s 2 + ∆ 2 − 1)z 1 · z 2 − (z 1 · k 1 )(z 2 · k 2 )W −− 12 + [(s 1 + ∆ 1 − 1)(k 2 · z 2 )(z 1 · K 12 ) + (1 ↔ 2)] (E.7) S ++ 23 and S ++ 13 are once again defined by cyclic permutations of (E.7). The operator H 12 which raises the spin at points 1 and 2 and also lowers the weight at both the points is given by : H 12 = 2 (z 1 · K 12 )(z 2 · K 12 ) − 2 (z 1 · z 2 )W −− 12 (E. 8) The operator that raises the spin at point 1 and simulataneously lowers the weight at point 2 is given by : A (1 ↔ 2) exchange in this operator gives D 21 . Both of these were used in the construction of T T O .

F Embedding space parity-odd correlation functions in four-dimensions
In this appendix, we show that JJT odd and T T T odd are zero in four-dimensions. Now, under simultaneous exchange of (Z 2 , Z 3 ) and (X 2 , X 3 ) Therefore, Hence, we must have A 001 = 0. Therefore, symmetry considerations force T T T odd to be zero.