Momentum space spinning correlators and higher spin equations in three dimensions

In this article, we explicitly compute three and four point momentum space correlation functions involving scalars and spinning operators in the free bosonic and the free fermionic theory in three dimensions. We also evaluate the five point function of scalars in the free bosonic theory. We discuss techniques which are more efficient than the usual PV reduction to evaluate one loop integrals. Our techniques may be easily generalized to momentum space correlators involving complicated spinning operators and to higher point functions. Three dimensional fermionic theory has an interesting feature that the scalar operator $\bar{\psi}\psi$ is odd under parity. To account for this feature, we develop a parity odd basis which is useful to write correlation functions involving spinning operators and an odd number of $\bar{\psi}\psi$ operators. We further study higher spin (HS) equations in momentum space which are algebraic in nature and hence simpler than their position space counterparts. We use HS equation to solve for specific three point functions involving spinning operators without invoking conformal invariance. However, at the level of four point functions we could only verify our explicit results. We observe that solving for the four point functions requires additional constraints that come from conformal invariance.


Introduction
Conformal field theory (CFT) plays a central role in theoretical physics. It has wide applicability ranging from very small length scales in particle physics, intermediate scales in condensed matter physics to large scales in cosmology. Recently with the development of the conformal bootstrap program, a lot of insight has been gained into the structure of CFT (see [1][2][3][4] and references therein). Most of the development in conformal bootstrap program is achieved in the position space whereas its momentum space analogue has not gained much attention until recently [5][6][7][8][9]. Three point momentum space correlation functions in CFTs have been determined through conformal invariance in [5,6,[10][11][12][13][14][15]. The absence of the analogue of conformal cross ratios u and v in momentum space makes the computation of the four and higher point correlation functions difficult. Furthermore, in general solving special conformal Ward identity is hard because these are complicated second order differential equations in momentum variables. For specific higher point correlators limited progress has been made recently in [16][17][18][19][20][21]. It is quite desirable to understand the CFT in momentum space for various reasons which include its relation to Feynman graphs which are usually computed in momentum space. It is also significant for its applicability in the context of cosmology [22][23][24][25][26][27][28][29]. Although there are difficulties involved in imposing conformal invariance in momentum space there are some attractive features as well. For example, conformal blocks become simple products of two three-point functions which follows from the fact that the descendants are related to the primary operators by a simple multiplication of the momenta.
In the present article, we determine explicit three and four point correlators of some spinning operators in three dimensional free bosonic and free fermionic theories. In three dimensions, there exist parity odd correlators in the fermionic theory. We develop a basis to express such parity odd three and four point correlators. In order to compute the scalar and spinning four point correlator, we use some existing techniques and also develop some efficient methods to evaluate one loop three dimensional integrals.
The other aspect that we explore in this paper is to understand the higher-spin equations [30][31][32][33][34][35] in momentum space. In momentum space, the higher spin equation are algebraic and hence, becomes particularly simple as compared to their counterparts in position space. We utilize this aspect to solve for some of the three point correlators involving spinning operators without invoking conformal invariance. However, solving the higher spin equation for four point correlators is more complicated. Hence in this article we only verify that the four point correlators we determine solve the higher spin equations.
Our article is organized as follows. In section 2 we list the theories and their corresponding action, spectrum and the explicit form of the higher spin operators which are of interest to us in the present article. In section 3 we describe the conformal Ward identities and the reconstruction formula for the three and four point spinning correlators we study. In section 4 we introduce the momentum basis for the transverse part of both parity odd and even higher spin correlators. In section 5 we provide explicit results for the three and four point correlators involving spinning and scalar operators in both free bosonic and free fermionic theories in three dimensions. In section 6 we determine the three point correla-tors by solving the higher spin equation. Furthermore we also verify that the four point functions we determined solve the corresponding higher spin equation. In Appendix A and B we provide details of the computation of some of the correlators.In particular we show how to utilize the Schouten identity and the inversion technique to evaluate integrals more efficiently than the usual PV reduction. In appendix C we describe some details of the higher spin equation.

List of Theories and Operators
In this section we describe the various theories which are of interest to us in the present article. We discuss their Lagrangians and operator spectrum here.

Free Bosonic Theory
The simplest of the conformal field theories we consider here involve the masless free bosonic theory in 2 + 1 dimensions. Let us consider the scalar to be in fundamental representation of SU (N ) 1 . The action for this theory is given as follows Note that this theory exhibits a higher spin symmetry which implies that it has a tower of exactly conserved higher spin currents which we denote as J s where-s represents the spin of the operator-J s . In this article we will mostly be dealing with s = 0, s = 1 and s = 2 operators which correspond to the scalar, spin one and spin-2(stress tensor) conserved current respectively which are as follows Note that we have chosen the stress tensor in such a way that it is explicitly traceless. As we will be dealing with momentum space mostly, we convert the above operators to momentum space to obtain the following expression for them Most of the discussion and the results for free bosons in this article are also applicable to the complex As is well known these form the single-trace primary operators of this conformal field theory and their scaling dimensions are given by Note that the above scaling dimensions do not receive any anomalous corrections as the currents are exactly conserved.

Free Fermionic theory
Having described the action and spectrum of the free bosonic theory, we now proceed to discuss the same for theory of masless free fermions in 2 + 1 dimensions which are once again in the fundamental representation of SU (N ). The action of this theory is as follows This conformal field theory (CFT) also consists of a tower of exactly conserved currents. These are the single trace primaries of the CFT, one for each spin-s. For s = 0, s = 1 and s = 2 operators which correspond to the scalar, spin one and spin-2(stress tensor) conserved current respectively are given as The above operators in the momentum space obtained through Fourier transform are as follows The scaling dimensions of these operators are given as Note that the spectrum of higher spin operators appearing in the free fermionic theory and free bosonic theory are same except for the scalar operators. Also, observe that the parity of the scalar operator in bosonic theory is even, where as in the fermionic theory it is odd.

Conformal Ward identities
In this section we describe in detail the conformal Ward identities obeyed by correlation functions involving scalar and spinning operators [6]. We then focus on correlation functions involving conserved currents and implement the constraints coming from diffeomorphism invariance (conservation laws) and the Weyl Ward identity (trace Ward identity).
We consider the n-point function of primary operators O 1 , . . . , O n in a CFT and denote it by O 1 (k 1 )O 2 (k 2 ) . . . O n (k n ) . Some of these operators might have a non-zero spin and hence have Lorentz indices, but we suppress them here for sake of brevity. Let the conformal dimension of O i be ∆ i . We denote the correlator with the momentum conserving delta function stripped off using double bra-kets as O 1 (k 1 ) O 2 (k 2 ) . . . O n (k n ) . Thus we have : We will now discuss the Ward identities that O 1 (k 1 ) . . . O n (k n ) has to satisfy.

Dilatation and Special conformal Ward identities
The dilatation Ward identity on a generic correlator with scalar and tensor insertions is given by [6] : This imposes the following scaling behaviour on the correlator : The special conformal Ward identity on the n point correlator of scalar primaries is [6] : When the correlator involves spinning operators the special conformal Ward identity is modified by an additional differential operator that mixes the tensor indices of the correlator. The additional term takes the following form [6] : For example, in three dimensions the dilatation and special conformal Ward identities on the J µ J ν J 0 correlator when the scalar operator J 0 has dimension 1 take the form : For the case of the three point correlator in the boson theory this gives, This can be readily extended to the four point case as follows : For general n-point functions (n ≥ 4) with two spin one currents, the transverse Ward identity takes the form : For the fermion theory, since the current takes the simple formψγ µ ψ the transverse Ward identity is trivial and gives : We will now discuss the Ward identities associated to Weyl invariance.

Weyl/Trace Ward identities
The trace Ward identity for a correlator involving stress-tensor and scalars with dimension ∆ is given by For the four point correlator this becomes :

Reconstruction formula
As noted earlier, correlation functions involving conserved currents can be written as the sum of a transverse part and a local part. The local part of these correlators is fully determined in terms of lower point functions. In this sub-section, we give explicit expressions for the local part of correlators involving either a stress-tensor insertion or two spin one insertions and scalars of dimension ∆ in d dimensions [20].
For the three point correlator with a stress-tensor insertion we have : where the local part is given by [20] : For the four point correlator the local part takes the following form [20] : For the three point correlator with two spin one insertions one can do a similar splitting as in (3.19) where in the bosonic theory For the four point correlator in the bosonic theory the local part is given by : In the fermionic theory owing to the fact that the transverse Ward identity is trivial (3.16) there is no contribution to correlators involving two spin one currents from the local part.
We will now turn our attention to the transverse part of the correlators. To express the transverse part compactly, we need to work with a suitable momentum basis. We discuss this in the following section.

Momentum basis to expand transverse part of the correlator
For correlator involving spinning operators, it is conveient to introduce suitable basis to express the results. In this section we describe the momentum space projector basis that could be utilized to write the transverse part of the correlators of various operators in the theories we described above. For parity even correlator, this is already dicussed in the work of [6]. However, we also introduce another basis for parity even correlators, which is particularly useful in three dimensions. For our discussion, we shall also require parity odd correlation functions. We also develop momentum basis to expand such correlators.

Projector basis for parity even correlator
The simplest of projector which projects orthogonal to momentum can be written as It can be easily seen that the above projector is orthogonal to p For a correlator involving stress tensor, one can use following projector Note that the above projector is orthogonal to p and is also traceless The other projector basis that appears in our computation of the correlator with two spinone conserved currents is as follows where π µ α (p 1 ) , π ν β (p 2 ) may be obtained from the definition given in eq.(4.1).Note that the above projector is orthogonal to p 1 in the µ index and orthogonal to p 2 in the ν index

A different basis involving Levi-Civita tensor in three diemsnions
Quite interestingly in 3d there exists an equivalent set of projectors to the above in terms of the Levi-Civita tensor. We notice that the simplest projector in (4.1) can be re-expresssed in terms of the Levi-Civita as follows The orthogonal and traceless projector equivalent to the one in (4.3) can be written as where we are using the following notation for compactness It is interesting to compare (4.8) with (4.3). With little algebra one can show that As earlier in 3 dimensions the projector equivalent to the one in eq.(4.5 ) which is orthogonal to p 1 in the µ index and orthogonal to p 2 in the ν index is expressed in terms of Levi-Civita tensor as follows This completes our discussion on parity even projectors. We now write basis for parity odd correlator.

Projector basis for parity odd correlators
In three dimension one expects presence of parity odd correlators. For this purpose, it is useful to descibe projector basis which one can use to describe such correlators. Let us note that, to obtain parity odd basis we can use Levi-Civita tensor. For example, the analougue of (4.3), (4.8) would be Note that with the above defination ∆ µν αβ (p) is automatically orthogonal and traceless i.e. and (4.14) One more useful basis that one can use is where p i in the last equation (4.15) can be either p 1 or p 2 . This basis is useful in describing three point function involving two spin one current J µ and one parity odd operator J F 0 .
In some situations, one can also define parity odd projector 2 by just multiplying parity even projector with an Levi-Civita tensor. For example, in the case of four point function one can define parity odd projector tensor as However, it is important to note that (4.16) may not be very convenient in the sense that results can look very complicated in this basis. For example, to describe the parity odd correlator for free fermion theory, it is conveient to use (4.12) rather than

Results of explicit computations
In this section we shall present explicit results for correlation functions of interest to us both in free bosonic and free fermionic theory in three dimensions.

Free bosonic theory in d = 3
In this section we write down results for two, three and four point function of various operators in free bosonic theory.

Two and Three point functions
The two point function of scalar operators J B 0 (k) is given by : The three point function of scalar operator J B 0 (k) is given by Let us now compute the simplest three point function with a spinning operator which is where J B µ corresponds to the conserved current in the bosonic theory given by (2.4). The Wick contractions explicitly show that the three point function is zero : This can be understood from the fact that J µ and hence the above correlator is charge conjugation odd. It is impossible to write a function of only the momenta which is charge conjugation odd. This argument readily generalises to the vanishing of an arbitrary n-point correlator with an odd number of spin one current operator in the free theory.
The first non-trivial higher spin correlator which is of interest to us is T B µν J B 0 J B 0 . The stress tensor of the free boson theory is as given in (2.2). Note that we choose to work with a stress-tensor which is explicitly traceless.

Tensor Decomposition: Local and Transverse parts
As discussed in (3.19) of Section 3.3, the correlator can be written as the sum of a transverse part and a traceless part. The transverse part is given by where the transverse-traceless projector Π µν αβ (p) is defined in (4.3). By performing the integrals from the Wick contractions we determined the form factor A 1 (k 1 , k 2 , k 3 ) : The expression we obtained for the form factor (5.5) matches the result obtained by solving conformal Ward identities in [6] . Since we chose to work with a stress tensor that is traceless the local part of the correlator has an additional term in (3.20) given by : Due to the additional term (5.6) that appears in the local part of the correlator, the transverse Ward identities (3.9) are modified by the following additional piece : The trace Ward identity (3.17) also gets modified to give a zero on the R.H.S since the stress-tensor has been chosen to be traceless. It may be easily verified that the above function A 1 (k 1 , k 2 , k 3 ) in (5.5) obeys the primary Ward identities derived in [6] Interestingly, the transverse part of the correlator in (5.13) in three dimension can be reexpressed in terms of a different projector composed of Levi-Civita tensors as follows : where χ µν αβ (k 1 ) corresponds to the projector defined in (4.8). In this basis, the form factor A 1 (k 1 , k 2 , k 3 ) is given by Let us note the following relation between the form factors A 1 and A 1 which readily follows from (4.10).
We now present the results for the correlator J µ J ν J B 0 . Details of the computation are given in Appendix B.

Tensor Decomposition: Local and Transverse parts
Our result obtained by explicit integration can be expressed as the sum of a transverse part and a local part as in (3.22) [6] : The transverse part of the correlator is given by where the projector π µ α (k 1 ) is defined in (4.1). Through explicit computation we determined the form factors turn to be : The local part of the correlator is given as in (3.23) which then satisfies the transverse Ward identities in (3.13). It is easy to check that the form factors A 1 and A 2 satisfy the primary Ward identities [6] : Interestingly, the transverse part of the correlator in (5.13) can be expressed in terms of a different projector composed of Levi-Civita tensors as follows : where A 1 (k 1 , k 2 , k 3 ) and A 2 (k 1 , k 2 , k 3 ) are given by :

Four point functions
We will now describe the results of our computation of various four point functions in the free boson theory. In order to evaluate the results, we utilize two interesting techniques. The first technique involves inversion of all the momenta appearing in the integral (See appendix.A.1 and section 5.3.2 of [36] and [37]). Through this method one may compute three dimensional box integrals in momentum space easily and efficiently. The second method is to incorporate the Schouten identity(See (A.10) of the appendix.A.3 and see also [38]) to simplify the integral appearing in the computation of the required correlators, into the known integrals. This method turns out be much more efficient for more complicated integrals than the usual PV reduction scheme.
The four point correlator of the scalar operator J B 0 is obtained by three relevant Wick contractions followed by the use of inversion technique to evaluate the integrals. Note that the scalar four point correlator was determined in a specific kinematic regime in [39,40]. The final result is given as follows In the above equation (2 ↔ 3) and (3 ↔ 4) denote that these terms are related to the first term by the momentum exchange k 2 ↔ k 3 and k 3 ↔ k 4 respectively.
Having computed the scalar correlator, we now proceed to compute the first non-trivial higher spin correlator We follow [20] closely. The computational details are given in Appendix B.1 The transverse part of the correlator is expressed in terms of the projector as follows : where the transverse-traceless projector Π µν αβ (k 1 ) is given by (4.3). Interestingly, exchange symmetry k 2 ↔ k 3 , k 2 ↔ k 4 and k 3 ↔ k 4 dictates that the form factor A is the only independent one of the three in (5.19). One may express other two form factors in terms of A as follows The above relations imply that we just need to evaluate one form factor A. Explicit computation gives where W µ and the b i are as follows :

Local Part and Ward Identity
As in the three point case the local part of the correlator (3.21) gets an additional contribution as we chose to work with a traceless stress-tensor. It is given by The transverse Ward identities are also modified accordingly with the additional terms being given by : where the scalar three point function is as given in (5.2).
As before the correlator can be expressed as a sum of transverse and local parts (3.22). The transverse part of the correlator could be expressed in terms of the projector as follows where ζ µν αβ is the projector defined in (4.5). Using exchange symmetry k 3 ↔ k 4 and µ ↔ ν, k 1 ↔ k 2 it is easy to show that D 1 and B 1 are determined in terms of C 1 as given below Thus the only independent form factors are A 1 and C 1 . Upon explicit computation, we determined these to be : where the vector W µ and b i 's are as given in (5.22). Interestingly, the form factor C 1 (k 1 , k 2 , k 3 , k 4 ) can be related to form factor A(k 1 , k 2 , k 3 , k 4 ) in (5.21) which arise in privious subsection as follows The local part of the correlator is as given in (3.24) and which satisfies the expected transverse Ward identities (3.14).

Five point functions
In this subsection we describe the result we obtained for the five point correlation of the scalar operator J B 0 in the free bosonic theory. We provide the details of computation in Appendix B.2.
Although the method of Inversion of momenta proves to be efficient for the computation of the scalar four point correlators, it turns out to be a bit tedious for the case of five point function. However the technique of utilizing the Schouten identity turns out to be much more efficient and insightful for evaluating the five point correlators as demonstrated in [38] which we have decribed in the Appendix B.2. The relevant contractions give the following result The details of the computation of the above integral are provided in the appendix. We describe the final result for the above integral here H(k 1 , k 2 , k 3 , k 4 , k 5 ) = F (k 1 , k 12 , k 123 , k 1234 ) (5.30) where the function F is expressed in therms of the integrals appearing in the lower point functions which we have denoted by E Note that in the above equation the second line implies that E 1234 is obtained by E 0ijk by replacing p i = p  41 . One may now obtain H(k 1 , k 12 , k 123 , k 1234 ) by using (5.31) and (5.30), and then substitute the result in (5.30) to determine full five point correlator . We emphasize here that the scalar five point function we obtained has several interesting applications. For example, this was an important ingredient in determining a certain beta function in [41] . We emphasize that the method we have utilized could also be generalized straightforwardly to compute for example the five point correlator T µν J 0 J 0 J 0 J 0 .

Free Fermionic theory in d = 3
Having completed the discussion on the results in free bosnic theory, we now turn our attention to the correlation functions of the free fermionic theory. One interesting difference here is the fact that scalar operator J F 0 defined in (2.10) is parity odd. Hence, the correlation functions we compute in this section are going to be structurally different than what we observed in the previous section. Below, we present the results for two, three and four point function of various operators in the free fermionic theory.

Two and Three point functions
The two point function of scalar operators J F 0 (k) is given by : The three point function of scalar operator J F 0 (k) can be easily shown to be vanishing The stress tensor for the fermionic theory is as given in (2.12). The tensor decomposition of the transverse-traceless part of the correlator can be expressed as By performing the integrals that appear in the correlator explicitly using the results given in appendix.A, we obtained the following expression for the form factor It can be easily checked that the form factor satisfies the dilatation Ward identity [6], and the primary Ward identities [6], The local part of the correlator is given by (3.20) for d = 3 and ∆ = 2. It satisfies the transverse and trace Ward identities in (3.9) and (3.17) respectively for for d = 3 and ∆ = 2.
The current J F µ for the free fermionic theory is as given in (2.12). Here we proceed to compute the three point correlator J F µ J F ν J F 0 . Given the transverse Ward identities (3.16) and the fact that the correlator is odd under parity we find it useful to write the correlator in the following form where Σ µν αβ is the momentum basis projector for the transverse part of the parity odd correlator we introduced in (4.11). The presence of an odd number of Levi-Civita tensors in

Four point functions
Having obtained the three point functions in the fermionic theory we now proceed to compute the four point correlation functions.
Let us begin with simplest four point function which is of the scalar operator J F 0 . Note that this scalar four point correlator was obtained in a specific kinematic regime in [39,40]. The relevant Wick contractions lead to the integrals of the forms that are provided in Appendix A. Upon utilizing them we obtain the following result for the correlator where the function χ(k 1 , k 2 , k 3 , k 4 ) is given as follows Unlike the corresponding four point correlation function in the free-bosonic theory, the free fermionic correlator has only the transverse part. This is because the local part (3.21) which is composed of the three point three point correlator of scalars J F 0 vanishes in the theory (5.2). Given the fact that T F µν J F 0 J F 0 J F 0 is parity odd, the entire correlator may be expressed in terms of parity odd projector in (4.12) as follows This is by construction explicitly transverse to momentum k 1 . We determined the form factors by explicitly performing the integrals appearing in the relevant Wick contractions. The required integrals are once again of the form given in appendix.A and we obtain the following expression for the coefficient B 2 B 2 (k 1 , k 2 , k 3 , k 4 ) = k 2 2k 12 k 123 (k 1 k 123 + k 12 k 13 + k 2 k 3 ) − k 2 (k 12 k 2 + k 123 k 23 ) 2k 12 k 123 k 23 k 3 (k 1 k 3 + k 12 k 23 + k 123 k 2 ) Note that vectorW µ and b i (i = 1, 2, 3)s are those which appeared in the corresponding correlator for the bosonic theory as given by (5.22). The form factors A 2 and C 2 are determined in terms of B 2 as follows This completes our computation of the required correlator

Five point function
The explicit computation of the trace identities of the gamma matrices could be used to demonstrate that the five point function of J F 0 operators in the free fermionic theory vanishes. We also emphasize that one may generalize our techniques straightforwardly to compute correlators of the sort

Correlators from higher spin Ward identities
In this section we discuss how correlators in theories with a higher spin symmetry can be obtained by solving the corresponding Ward identities [30][31][32][33][34][35]. If we denote the charge associated to the higher spin symmetry by Q s 3 , the Ward identity takes the form : Higher spin equations in momentum space have the nice feature that they are algebraic as opposed to differential equations in position space. Hence, one would expect them to be easily solvable. However, the fact that we do not know the analogue of the position space conformal cross ratios u and v in momentum space leads to a proliferation of unknown variables to solve for which most often exceed the number of algebraic constraints coming from the higher spin equations. In such cases one needs to resort to conformal invariance to get the additional constraints required to solve for the correlators. Nevertheless, we shall see that we are able to solve for some three point functions without invoking conformal invariance. At the level of four point functions we show that higher-spin equations can be used to verify the explicit results for correlators which we obtained by direct computation.
To illustrate our approach we shall start with the bosonic theory and then move on to the fermionic theory.

Three point spinning correlators from higher spin equations
In this section we discuss how the higher spin Ward identities may be put to use to compute some three point spinning correlators in the free bosonic theory without invoking conformal invariance. In our analysis we stick to spin 2 and spin 3 charges denoted by Q 3 and Q 4 respectively. The action of these charges on the spin 0 and spin 1 operators is given by : One can check that the algebra is consistent with the data in Table. 1. Especially one can see that although dimension and spin would allow for a stress-tensor term in the [Q 3 , J 0 ] algebra, charge conjugation forbids this. Note that in this section we drop the superscripts B and F that distinguish operators in the bosonic theory and the fermionic theory. This shall be clear from the context.

Operator
Dimension (∆) Spin (s) Twist (τ ) Charge conjugation Parity Q 3 2 2 0 odd even Q 4 3 3 0 even even J 0 1 0 1 even even J 1 2 1 1 odd even T µν 3 2 1 even even Table 1: Data of charges and operators of interest in the bosonic theory Computing T J 0 J 0 from higher spin equations We will first show how to obtain the T J 0 J 0 correlator by solving a higher spin equation. The first step in this regard is to identify the higher spin charge and the correlator on which the chosen charge must act. From the action of the charges in (6.2) it is clear that the Ward identity (6.1) for the charge Q 4 on the correlator J 0 J 0 J 0 generates T J 0 J 0 . The higher spin Ward identity takes the form : where in the second line we used the algebra in (6.2). We now express this equation in the momentum space by performing a Fourier transform : where the local part of the T µν J 0 J 0 correlator is given by the sum of the expressions in (3.20) and (5.6). Our goal is to solve (6.4) for the form factor in the transverse-traceless part : Accounting for the three form factors that appear in the second line of the R.H.S of (6.4) (one from each of the three correlators) solving (6.4) amounts to solving three linearly independent equations in three variables. Solving them we obtain the following for the form factor in the T µν J 0 J 0 correlator : The form factors in J 0 T µν J 0 and J 0 J 0 T µν are also solved for and they are related to A(k 1 , k 2 , k 3 ) in (6.6) by k 1 ↔ k 2 and k 1 ↔ k 3 exchange, respectively. This matches the result obtained by direct computation in (5.5).
Computing J 1 J 1 J 0 from higher spin equations Now that we have solved for the form factor in the T µν J 0 J 0 correlator, we can use it to solve a higher spin Ward identity to obtain the form factors in the transverse part of the J µ J ν J 0 correlator : From (6.2) it is clear that the Ward identity (6.1) for the charge Q 3 on the correlator J − (k 1 )J 0 (k 2 )J 0 (k 3 ) generates the required correlator. Following (6.3) and (6.4) the higher spin equation takes the following form in momentum space : Having solved for the form factor in the transverse-traceless part of the T µν J 0 J 0 correlator (6.6), one can combine it with the local term in T µν J 0 J 0 (see (3.20), (5.6)) and the scalar three point function (5.2) to solve (6.8) for the form factors A 1 , A 2 , A 3 and A 4 in (6.7). Note that the J µ J ν J 0 correlator has no local term. The higher spin equation (6.8) gives four linearly independent equations in these four unknowns. The equations can be easily solved to obtain : .
The form factors A 3 (k 1 , k 2 , k 3 ) and A 4 (k 1 , k 2 , k 3 ) are also solved for and are given by a k 2 ↔ k 3 exchange in A 1 (k 1 , k 2 , k 3 ) and A 2 (k 1 , k 2 , k 3 ) respectively. This precisely matches the results obtained in (5.14).
With the analysis of this subsection, we have illustrated how the higher spin equations determine the three point spinning correlators in the bosonic theory without invoking conformal invariance.

Four point spinning correlators and higher spin equations
In this section we extend the analysis of the previous section to 4 point functions. Let us first look at the Ward identity corresponding to the charge Q 4 on the J 0 J 0 J 0 J 0 correlator. Following (6.3) and (6.4) the higher spin equation takes the following form in momentum space : The transverse-traceless part of the correlator is given by Given the higher spin equation (6.10) it can be checked that the scalar four point function (5.18) and the spinning correlator J 2 J 0 J 0 J 0 computed in section 5.1.2 solve it. However, it is not possible to solve for the form factors in T J 0 J 0 J 0 as the above is a set of 10 linear equations in 12 variables.
We will now show how the form factors in the T µν J 0 J 0 J 0 correlator can be solved for with the additional knowledge of the spinning correlator J µ J ν J 0 J 0 . For this we consider the Ward identity corresponding to the action of Q 3 on J − J 0 J 0 J 0 : One can easily verify that the explicit results obtained for the scalar four point correlator in (5.18), the T µν J 0 J 0 J 0 correlator in section 5.1.2 and the J µ J ν J 0 J 0 correlator in section 5.1.2 solve the higher spin equation (6.12). But one can do better by actually solving for the form factors. From (6.12) it is clear that given the spinning correlator J µ J ν J 0 J 0 and the four point function of the scalars, the higher spin Ward identity completely determines the form factors A, B, and C that appear in the transverse-traceless part of the T µν J 0 J 0 J 0 correlator (6.11). However, the converse is not quite true. Solving for the form factors in J µ J ν J 0 J 0 given the T µν J 0 J 0 J 0 correlator amounts to solving for 12 variables in 10 linear equations.
In cases where the number of unknowns to solve for exceed the number of higher spin equations, one requires additional input and this comes from the constraints imposed by conformal Ward identities. In the following sub-section we illustrate this in the context of the free fermion theory where one can see this already at the level of three point functions.

Free fermion theory
In this sub-section we will discuss how the three point correlator T µν J 0 J 0 in the free fermion theory may be obtained by solving the higher spin Ward identity. For this we impose the Ward identity (6.1) corresponding to the generator Q 4 on the correlator J 0 (k 1 )J 0 (k 2 )J 0 (k 3 ) . The action of Q 4 on J 0 is given by : Combined with (6.1) we get the following higher spin equation in momentum space : Operator Dimension (∆) Spin (s) Twist (τ ) Charge conjugation Parity Q 4 3 3 0 even even J 0 2 0 2 even odd J 1 2 1 1 odd even T µν 3 2 1 even even Since the three point J 0 (k 1 )J 0 (k 2 )J 0 (k 3 ) vanishes in the free fermion theory, the equation becomes One can easily check that the expression for T µν J 0 J 0 (see (3.20) and section 5.2.1) solves (6.15). We now ask if we can do better and actually solve for the form factor that appears in the transverse-traceless part of the T µν J 0 J 0 correlator (5.36) using the higher spin equation (6.15). A simple counting of the number of unknowns and independent equations makes it clear that the higher spin equation is not powerful enough to solve for all the form factors individually. Solving for two of the form factors A 2 (k 1 , k 2 , k 3 ) and A 3 (k 1 , k 2 , k 3 ) that appear in T µν (k 2 ) J 0 (k 1 ) J 0 (k 3 ) and T µν (k 3 ) J 0 (k 1 ) J 0 (k 2 ) respectively in terms of the form factor in T µν (k 1 ) J 0 (k 2 ) J 0 (k 3 ), viz. A 1 (k 1 , k 2 , k 3 ) we obtain One can easily verify that the form factors obtained by an explicit computation in (5.37) satisfy the above relations. However, to completely determine the form factors we require one more constraint and that is provided by imposing conformal invariance. It can be easily checked that given A 1 satisfies the dilatation Ward identity, A 2 (A 1 ) and A 3 (A 1 ) in (6.16) satisfy the identity trivially, i.e. it does not give us any new constraint. Therefore to solve for the form factors, we combine the higher spin equation with the primary Ward identities. After a series of steps that involve imposing conformal invariance one arrives at the following first order differential equation : For details of this computation and the explicit form of g 1 (k 1 , k 2 , k 3 ), g 2 (k 1 , k 2 , k 3 ) and g 4 (k 1 , k 2 , k 3 ) see Appendix C. Since this is a first order differential equation this can in principle be solved. Here we note that the form factor A 1 (k 1 , k 2 , k 3 ) obtained in (5.37) satisfies this first order differential equation. Thus the three point correlation function T µν J 0 J 0 in the fermionic theory is determined by the higher spin equations aided by conformal invariance where the latter is imposed through a first order differential equation.

Discussion
In this paper we have explicitly computed various parity odd as well as parity even correlation functions involving scalar and spinning operators in three dimensional free theories. In particular we developed a basis for the transverse part of the parity odd correlators. Furthermore we demonstrated that the techniques involving Schouten identity and inversion method provide an efficient way to compute the correlators. We then explored the higher spin equations for free theories. We demonstrated that some of the three point functions could be solved solely using the higher spin equations. However we could only verify that the four point function solves the higher spin equation.
There are a few immediate generalisation of our work that one can pursue. Generalising these results to the interacting theories such as the Chern-Simons matter theories would be interesting [42]. One may use slightly broken higher spin symmetry to solve for correlator for those cases. It would also be exciting to find out structures which are not dictated by free theory, using the higher spin equations. Another interesting question is to ask if higher spin equation can be used to understand contact terms which appear in momentum space quite often. It would also be interesting to explore the double copy structure of the CFT correlators such as [43,44]. Another interesting direction to pursue is to understand the spin and weight raising operators of [27][28][29] for parity odd correlators. This is not immediately clear from [27][28][29] since the seed three point function of scalars in the fermionic theory is zero.
It would be interesting to examine higher spin equations in the context of [45]. In the case of four point correlators, we saw that although explicit results solved the higher spin equation, one could not solve the equations to obtain the correlators. This was because there were more variables to solve for than equations. The large number of variables stemmed from a lack of understanding of conformal cross-ratios in momentum space. We believe that the recent work [45] gives a way to deal with this issue. We hope to come back to these issues in the near future.
Here we give the details of the computation of the integral required to compute the five point function of the scalar operator J B 0 in free bosonic theory. The method of inversion of the momentas however for this computation is a bit tedious. Hence here we employ the method of determining the integral required to compute the five point function in terms of the integrals appearing in the four point function through an interesting identity known as the Schouten Identity as suggested by authors in [38]. The Schouten identity for a three dimensional vector is as follows p 1 p 2 p 3 l µ = µp 2 p 3 l.p 1 + p 1 µp 3 l.p 2 + p 1 p 2 µ l.p 3 (B.11) As explained in [38], the presence of another independent momentum p 4 simplifies the computation of the integral 4 as one may construct a linear equation by dotting the above identity with the momentum p 4 . This leads to the following l p 1 p 2 p 3 l.p 4 − ( p 4 p 2 p 3 l.p 1 + p 1 p 4 p 3 l.p 2 + p 1 p 2 p 4 l.p 3 ) l 2 (l + p 1 ) 2 (l + p 2 ) 2 (l + p 3 ) 2 (l + p 4 ) 2 = 0 (B.12) We may now express l.p i = 1 2 ((l + p i ) 2 − l 2 − p 2 i ) and reduce most of the above integrals to the integrals that appear in the computation of four point correlation function. This leads to an interesting relation upon simplification which can be expressed in a compact notation as follows F (p 1 , p 2 , p 3 , p 4 , p 5 ) = l 1 l 2 (l + p 1 ) 2 (l + p 2 ) 2 (l + p 3 ) 2 (l + p 4 ) 2 4 One can also perform a similar computation to determine the integral that appears in the computation of four point from the integrals for three point function. However in this case the inversion method we utilized is more efficient and simple. See [38] for details. where F (k 1 , k 12 , k 123 , k 1234 ) can be obtained from eq.(B.13) by the integrals that appear in the four point correlator.

C Details of the higher spin equation
The primary Ward identity on the form factors is given by : K ij A 1,2,3 = 0, i, j = 1, 2, 3 (C.1) where K ij = K i − K j and Given A 2 as a function of A 1 (6.16) we impose (C.1) by acting K 12 on A 2 (A 1 ) : We now impose in (C.3) the following primary Ward identity that A 1 must satisfy : This reduces the second order differential equation to a first order differential equation : g 1 (k 1 , k 2 , k 3 ) + g 2 (k 1 , k 2 , k 3 )A 1 + g 3 (k 1 , k 2 , k 3 ) ∂A 1 ∂k 1 + g 4 (k 1 , k 2 , k 3 ) ∂A 1 ∂k 2 = 0 (C.5) where