Three-point functions of conserved currents in 4D CFT: general formalism for arbitrary spins

We analyse the general structure of the three-point functions involving conserved higher-spin currents $J_{s} := J_{\alpha(i) \dot{\alpha}(j)}$ belonging to any Lorentz representation in four-dimensional conformal field theory. Using the constraints of conformal symmetry and conservation equations, we computationally analyse the general structure of three-point functions $\langle J^{}_{s_{1}} J'_{s_{2}} J''_{s_{3}} \rangle$ for arbitrary spins and propose a classification of the results. For bosonic vector-like currents with $i=j$, it is known that the number of independent conserved structures is $2 \min (s_{i}) + 1$. For the three-point functions of conserved currents with arbitrarily many dotted and undotted indices, we show that in many cases the number of structures deviates from $2 \min (s_{i}) + 1$.


Introduction
In conformal field theory (CFT), it is known that the general structure of threepoint functions of conserved currents is highly constrained by conformal symmetry.A systematic approach to study three-point functions of primary operators was introduced in [1,2] (see also refs.[3][4][5][6][7][8][9][10][11][12] for earlier works), which presented an analysis of the general structure of three-point functions involving the energy-momentum tensor and conserved vector currents.The analysis of three-point functions of conserved higher-spin bosonic currents was later undertaken by Stanev [13][14][15] (see also [16,17]) in the four-dimensional case, and by Zhiboedov [18] in general dimensions. 1 In four dimensions (4D) it was shown that the number of independent structures in the three-point function of conserved bosonic vector-like currents J µ 1 ...µs increases linearly with the minimum spin.This is quite different to the results found in three dimensions (3D), where it has been shown by many authors [37][38][39][40][41][42][43] that there are only three possible independent conserved structures for currents of arbitrary integer/half-integer spins.The aim of this paper is to study threepoint functions of conserved currents belonging to arbitrary Lorentz representations in 4D CFT.An approach to this problem was outlined in [16], however, it did not study correlation functions when the operators are all conserved currents.
The main consequence of the notation (1.2) for the currents is that there are essentially only two types of three-point functions to consider: Any other possible three-point functions are equivalent to these up to permutations of the points or complex conjugation.The main aim of this paper is to develop a general formalism to study the structure of the three-point correlation functions (1.3), where we assume only the constraints imposed by conformal symmetry and conservation equations.In doing so we essentially provide a complete classification of all possible conserved threepoint functions in 4D CFT.The three-point functions of currents with q = 0, 1 have been studied in e.g.[13,[16][17][18].For bosonic conserved currents (q i = 0), it is known that threepoint functions of conserved currents with spins s 1 , s 2 , s 3 are fixed up to 2 min(s 1 , s 2 , s 3 )+1 solutions in general.We show that the same result also holds for three-point functions involving conserved currents with q = 1.The three-point functions of currents J (s,q) with q ≥ 2, however, are relatively unexplored in the literature.Conserved currents with q ≥ 2 naturally arise in superconformal field theories in four-dimensions.As an example, consider a N = 2 superconformal field theory possessing a conserved higherspin supercurrent, J α(s) α(s) with s ≥ 1, satisfying the following superfield conservation equation [45]: where D β i is the spinor covariant derivative in N = 2 superspace, and i = 1, 2 is an iso-spinor index.The component structure of these supercurrents was elucidated in [46,47].The N = 2 supercurrent J α(s) α(s) can be decomposed into the following collection of independent conformal N = 1 supercurrent multiplets (see [46,47] for more details): J α(s) α(s) , J α(s+1) α(s) , J α(s+1) α(s+1) . (1.5) These N = 1 supercurrents, in turn, contain a multiplet of conserved component currents [48].In particular, the N = 1 supercurrent, J α(s+1) α(s) ,2 contains a conserved component current, S α(s+2) α(s) , defined as follows: where implicit symmetrisation among all α-indices is assumed.Hence, the N = 2 higherspin supercurrent J α(s) α(s) contains a conserved component current S α(s+2) α(s) , which corresponds to q = 2 in our convention above.The N = 2 supercurrents have been constructed explicitly for the free hypermultiplet and vector multiplet in [46,47].
The formalism, which augments the approach of [1] with auxiliary spinors, is suitable for constructing three-point functions of (conserved) primary operators in any Lorentz representation.Our approach is exhaustive in the sense that we construct all possible structures for the three-point function consistent with the conformal properties of the fields.We then systematically extract the linearly independent structures and impose the constraints arising from conservation equations, reality conditions, properties under inversion, and symmetries under permutations of spacetime points.The calculations are automated for arbitrary spins, and as a result we obtain the three-point function in an explicit form which can be presented up to our computational limit, s i = 10.However, this limit is sufficient to propose a general classification of the results.
We would like to point out that though the formalism developed in this paper is conceptually similar to the one developed for three-dimensional CFT in [43], there are two important differences.First, in three dimensions, three-point functions of conserved currents can have at most three independent structures (two parity-even and one parity-odd), whereas in four dimensions the number of conserved structures (generically) grows linearly with the minimum spin.Second, for three-point functions in 3D CFT an important role is played by the triangle inequalities For three-point functions involving conserved currents which are within the triangle inequalities, there are two parity-even solutions and one parity-odd solution.However, if any of the triangle inequalities are not satisfied then the parity-odd solution is incompatible with conservation equations [37][38][39][40][41][42][43].This statement has been proven in the light-cone limit in [38,39] (see also [40] for results in momentum space).However, we found that in 4D CFT the triangle inequalities appear to have no significance.
The content of this paper is organised as follows.In section 2 we review the properties of the conformal building blocks used to construct correlation functions of primary operators in four dimensions.We then develop the formalism to construct three-point functions of primary operators of the form J α(i) α(j) , where we demonstrate how to impose all constraints arising from conservation equations, reality conditions and point switch symmetries.In particular, we utilise an index-free auxiliary spinor formalism to construct a generating function for the three-point functions, and we outline the pertinent aspects of our computational approach.In section 3, we demonstrate our formalism by analysing the structure of three-point functions involving conserved vector currents, "supersymmetrylike" currents and the energy-momentum tensor.We reproduce the known results previously found in [1,13,18].We then expand our discussion to include three-point functions of higher-spin currents belonging to any Lorentz representation, and provide a classification of the results.For this the structure of the solutions is more easily identified by using the notation J (s,q) , J(s,q) for the currents as outlined above.In particular, we show that special attention is required for three-point functions of the form J (s 1 ,q) J (s 2 ,q) J (s 3 ,0) with q ≥ 2. In this case the formula for the number of independent conserved structures is found to be quite non-trivial.The appendix A is devoted to mathematical conventions and various useful identities.In appendix B we provide some examples of the three-point functions J (s 1 ,q) J (s 2 ,q) J (s 3 ,0) for which the number of independent conserved structures differs from 2 min(s 1 , s 2 , s 3 ) + 1.Then, as a consistency check, in appendix C we provide further examples of three-point functions involving scalars, spinors and conserved currents to compare against the results in [16].

Conformal building blocks
In this section we will review the group theoretic formalism used to compute correlation functions of primary operators in four dimensional conformal field theories.For a more detailed introduction to the formalism as applied to correlation functions of bosonic primary fields see [1].Our 4D conventions and notation are outlined in appendix A.

Two-point functions
Consider 4D Minkowski space M 1,3 , parameterised by coordinates x m , where m = 0, 1, 2, 3 are Lorentz indices.For any two points, x 1 , x 2 , we construct the covariant twopoint functions (2.1) The two-point functions can be converted to spinor notation using the conventions outlined in appendix A: In this form the two-point functions possess the following useful properties: Hence, we find . (2.4) We also introduce the normalised two-point functions, denoted by x12 , From here we can now construct an operator analogous to the conformal inversion tensor acting on the space of symmetric traceless tensors of arbitrary rank.Given a two-point function, x, we define the operator along with its inverse The spinor indices may be raised and lowered using the standard conventions as follows: Now due to the property we have the following useful relations: (2.10b) The objects (2.6), (2.7) prove to be essential in the construction of correlation functions involving primary operators of arbitrary spins.Indeed, the vector representation of the inversion tensor may be recovered in terms of the spinor two-point functions as follows: Tr(σ m x σn x) . (2.11) Now let Φ A be a primary field with dimension ∆, where A denotes a collection of Lorentz spinor indices.The two-point correlation function of Φ A and its conjugate Φ Ā is fixed by conformal symmetry to the form where I is an appropriate representation of the inversion tensor and c is a constant complex parameter.The denominator of the two-point function is determined by the conformal dimension of Φ A , which guarantees that the correlation function transforms with the appropriate weight under scale transformations.

Three-point functions
Given three distinct points in Minkowski space, x i , with i = 1, 2, 3, we define conformally covariant three-point functions in terms of the two-point functions as in [1] where (i, j, k) is a cyclic permutation of (1, 2, 3).For example, we have

23
I mn (x 23 ) . (2.16) The three-point functions also may be represented in spinor notation as follows: (2.17) These objects satisfy properties similar to the two-point functions (2.3).Indeed, it is convenient to define the normalised three-point functions, Xij , and the inverses, ( Now given an arbitrary three-point building block X, it is also useful to construct the following higher-spin operator: along with its inverse (2.20) These operators have properties similar to the two-point higher-spin inversion operators (2.6), (2.7).There are also some useful algebraic identities relating the two-and threepoint functions at various points, such as (2.21)These identities are analogous to (2.15a), (2.15b), and admit generalisations to higherspins, for example (2.22) In addition, similar to (2.16), there are also the following useful identities: These identities allow us to account for the fact that correlation functions of primary fields can obey differential constraints which can arise due to conservation equations.Indeed, given a tensor field T A (X), there are the following differential identities which arise as a consequence of (2.23): (2.24b) Now let Φ, Ψ, Π be primary fields with scale dimensions ∆ 1 , ∆ 2 and ∆ 3 respectively.The three-point function may be constructed using the general ansatz where the tensor H Ā1 Ā2 A 3 encodes all information about the correlation function, and is constrained by the conformal symmetry as follows: (i) Under scale transformations of Minkowski space x m → x m = λ −2 x m , the threepoint building blocks transform as X m → X m = λ 2 X m .As a consequence, the correlation function transforms as which implies that H obeys the scaling property This guarantees that the correlation function transforms correctly under scale transformations.
(ii) If any of the fields Φ, Ψ, Π obey differential equations, such as conservation equations, then the tensor H is also constrained by differential equations.Such constraints may be derived with the aid of identities (2.24a), (2.24b).
(iii) If any (or all) of the operators Φ, Ψ, Π coincide, the correlation function possesses symmetries under permutations of spacetime points, e.g.
where (Φ) is the Grassmann parity of Φ.As a consequence, the tensor H obeys constraints which will be referred to as "point-switch symmetries".A similar relation may also be derived for two fields which are related by complex conjugation.
The constraints above fix the functional form of H (and therefore the correlation function) up to finitely many independent parameters.Hence, using the general formula (2.29), the problem of computing three-point correlation functions is reduced to deriving the general structure of the tensor H subject to the above constraints.

Comments on differential constraints
For three-point functions of conserved currents, we must impose conservation on all three space-time points.For x 1 and x 2 , this process is simple due to the identities (2.24a), (2.24b), and the resulting conservation equations become equivalent to simple differential constraints on H.However, conservation on x 3 is more challenging due to a lack of useful identities analogous to (2.24a), (2.24b) for x 3 .To correctly impose conservation on x 3 , consider the correlation function Φ

.29)
We now reformulate the ansatz with Π at the front These two correlators are the same up to an overall sign due to Grassmann parity.Equating the two ansatz above yields the following relation: (2.31) Now suppose H(X) (with indices suppressed) is composed of finitely many linearly independent tensor structures, P i (X), i.e H(X) = i a i P i (X) where a i are constant complex parameters.We define H(X) = i āi Pi (X), the conjugate of H, and also H c (X) = i a i Pi (X), which we denote as the complement of H.As a consequence of (2.21), the following relation holds: After inverting this identity and substituting it directly into (2.31),we apply (2.21) to obtain an equation relating H c and H Conservation on x 3 may now be imposed by using (2.24a), with x 1 ↔ x 3 .In principle, this procedure can be carried out for any configuration of the fields.
If we now consider the correlation function of three conserved primaries where ∆ i = s i + 2. The constraints on H are then as follows: (i) Homogeneity: Recall that H is a homogeneous tensor field satisfying where Ĥα(j 1 ) α(i 1 )β(j 2 ) β(i 2 )γ(i 3 ) γ(j 3 ) (X) is homogeneous degree 0 in X, i.e.

(iv) Reality condition:
If the fields in the correlation function belong to the (s, s) representation, then the three-point function must satisfy the reality condition Similarly, if the fields at J, J at x 1 and x 2 respectively possess the same spin and are conjugate to each other, i.e.J = J, we must impose a combined reality/pointswitch condition using the following constraint where (J) is the Grassmann parity of J.
Working with the tensor formalism is quite messy and complicated in general, hence, to simplify the analysis we will utilise auxiliary spinors to carry out the computations.

Generating function formalism
Analogous to the approach of [43] we utilise auxiliary spinors to streamline the calculations.Consider a general spin-tensor H A 1 A 2 A 3 (X), where A 1 = {α(i 1 ), α(j 1 )}, A 2 = {β(i 2 ), β(j 2 )}, A 3 = {γ(i 3 ), γ(j 3 )} represent sets of totally symmetric spinor indices associated with the fields at points x 1 , x 2 and x 3 respectively.We introduce sets of commuting auxiliary spinors for each point; U = {u, ū} at x 1 , V = {v, v} at x 2 , and W = {w, w} at x 3 , where the spinors satisfy Now if we define the objects then the generating polynomial for H is constructed as follows: The tensor H can then be extracted from the polynomial by acting on it with the following partial derivative operators: The tensor H is then extracted from the polynomial as follows: (2.48) The polynomial H, (2.46), is now constructed out of scalar combinations of X, and the auxiliary spinors U , V and W with the appropriate homogeneity.Such a polynomial can be constructed out of the following monomials: ) ) To construct linearly independent structures for a given three-point function, one must also take into account the following linear dependence relations between the monomials: ) ) ) These allow elimination of the combinations There are also the following relations involving triple products: which allow for elimination of the products P i Pj Pk , Pi P j P k , Pi P j Qk .These relations (which appear to be exhaustive) are sufficient to reduce any set of structures in a given three-point function to a linearly independent set.
The task now is to construct a complete list of possible (linearly independent) solutions for the polynomial H for a given set of spins.This process is simplified by introducing a generating function, F(X; U, V, W | Γ), defined as follows: where the non-negative integers, Γ = i∈{1,2,3} {k i , ki , l i , li , r i }, are solutions to the following linear system: ) ) (2.57c) Here i 1 , i 2 , i 3 , j 1 , j 2 , j 3 are fixed integers corresponding to the spin representations of the fields in the three-point function.From here it is convenient to define (2.58) Using the system of equations (2.57), we obtain in addition to ) k1 + k2 + k3 ≤ min(j 1 + j 2 , j 1 + j 3 , j 2 + j 3 ) . (2.60b) Hence, the conditions for a given three-point function to be non-vanishing are Indeed, this is the same condition found in [16].Now given a finite number of solutions Γ I , I = 1, ..., N to (2.57) for a particular choice of i 1 , i 2 , i 3 , j 1 , j 2 , j 3 , the most general ansatz for the polynomial H in (2.46) is as follows: where a I are a set of complex constants.Hence, constructing the most general ansatz for the generating polynomial H is now equivalent to finding all non-negative integer solutions Γ I of (2.57).Once this ansatz has been obtained, the linearly independent structures can be found by systematically applying the linear dependence relations (2.50)-(2.55).
Let us now recast the constraints on the three-point function into the auxiliary spinor formalism.Recalling that s 1 = 1 2 (i 1 + j 1 ), s 2 = 1 2 (i 2 + j 2 ), s 3 = 1 2 (i 3 + j 3 ), first we define: ) where, to simplify notation, we denote J (s,q) ≡ J s .The general ansatz can be converted easily into the auxiliary spinor formalism, and is of the form where ∆ i = s i + 2. The generating polynomial, H(X; U, V, W ), is defined as where is the inversion operator acting on polynomials degree (i, j) in ( ũ, ũ).It should also be noted that Ũ has index structure conjugate to U .Sometimes we will omit the indices (i, j) to streamline the notation.After converting the constraints summarised in the previous subsection into the auxiliary spinor formalism, we obtain: (i) Homogeneity: Recall that H is a homogeneous polynomial satisfying the following scaling property: where we have used the notation U (i 1 , j 1 ), V (i 2 , j 2 ), W (i 3 , j 3 ) to keep track of homogeneity in the auxiliary spinors (u, ū), (v, v) and (w, w).For compactness we will suppress the homogeneities of the auxiliary spinors in the results.
(ii) Differential constraints: First, define the following three differential operators: (2.68) Conservation on all three points may be imposed using the following constraints: Conservation at x 3 : where, in the auxiliary spinor formalism, H is computed as follows: Using the properties of the inversion tensor, it can be shown that this transformation is equivalent to the following replacement rules for the building blocks: (2.71e) (iii) Point switch symmetries: If the fields J and J coincide (hence i 1 = i 2 , j 1 = j 2 ), then we obtain the following point-switch constraint where (J) is the Grassmann parity of J. Similarly, if the fields J and J coincide (hence i 1 = i 3 , j 1 = j 3 ) then we obtain the constraint (2.74) Similarly, if the fields at J, J at x 1 and x 2 respectively possess the same spin and are conjugate to each other, i.e.J = J, we must impose a combined reality/pointswitch condition using the following constraint where (J) is the Grassmann parity of J.

Inversion transformation
In general, whenever parity is a symmetry of a CFT, so too is invariance under inversions.An inversion transformation I maps fields in the (i, j) representation onto fields in the complex conjugate representation, (j, i). 3 Hence, inversions map correlation functions of fields onto correlation functions of their complex conjugate fields.In particular, if the fields in a given three-point function belong to the (s, s) representation then it is possible to construct linear combinations of structures for the three-point function which are eigenfunctions of the inversion operator.We denote these as parity-even and parity-odd solutions respectively.Indeed, given a tensor (2.76) Hence, we notice that under I, H transforms into the complex conjugate representation.An analogous formula can be derived using the auxiliary spinor formalism.Given a polynomial H(X; U, V, W ) = X ∆ 3 −∆ 2 −∆ 1 Ĥ(X; U, V, W ), the following holds: It is easy to understand this formula as the monomials (2.49) have simple transformation properties under I: with analogous rules applying for the building blocks Pi , Qi .Since, for primary fields in the (s, s) representation the three-point maps onto itself under inversion, it is possible to classify the parity-even and parity-odd structures in H using (2.77).By letting Ĥ(X) = Ĥ(+) (X) + Ĥ(−) (X), we have (2.79) Structures satisfying the above property are defined as parity-even/odd for overall sign +/−.This is essentially the same approach used to classify parity-even and parity-odd three-point functions in 3D CFT [43], which proves to be equivalent to the classification based on the absence/presence of the Levi-Civita pseudo-tensor.However, it is crucial to note that in three dimensions the linearly independent basic monomial structures comprising H are naturally eigenfunctions of the inversion operator.The same is not necessarily true for three-point functions in four dimensions due to (2.78), as the monomials (2.49) now map onto their complex conjugates.Hence, we are required to take non-trivial linear combinations of the basic structures and use the linear dependence relations (2.50)-(2.55) to form eigenfunctions of the inversion operator.Our classification of parity-even/odd solutions obtained this way is in complete agreement with the results found in [13].
3 Three-point functions of conserved currents In the next subsections we analyse the structure of three-point functions involving conserved currents in 4D CFT.We classify, using computational methods, all possible three-point functions involving the conserved currents J (s,q) , J(s,q) for s i ≤ 10.In particular, we determine the general structure and the number of independent solutions present in the three-point functions (1.3).As pointed out in the introduction, the number of independent conserved structures generically grows linearly with the minimum spin and the solution for the function H(X; U, V, W ) quickly becomes too long and complicated even for relatively low spins.Thus, although our method allows us to find H(X; U, V, W ) in a very explicit form for arbitrary spins (limited only by computer power), we find it practical to present the solutions when there is a small number of structures.Such examples involving low spins are discussed in subsection 3.1.In subsection 3.2 we state the classification for arbitrary spins.Some additional examples are presented in appendix B.

Conserved low-spin currents
We begin our analysis by considering correlation functions involving conserved low-spin currents such as the energy-momentum tensor, vector current, and "supersymmetry-like" currents in 4D CFT.Many of these results are known throughout the literature, but we derive them again here to demonstrate our approach.

Energy-momentum tensor and vector current correlators
The fundamental bosonic conserved currents in any conformal field theory are the conserved vector current, V m , and the symmetric, traceless energy-momentum tensor, T mn .The vector current has scale dimension ∆ V = 3 and satisfies ∂ m V m = 0, while the energy-momentum tensor has scale dimension ∆ T = 4 and satisfies the conservation equation ∂ m T mn = 0. Converting to spinor notation using the conventions outlined in appendix A, we have: These objects possess fundamental information associated with internal and spacetime symmetries, hence, their three-point functions are of great importance.The possible three-point functions involving the conserved vector current and the energy-momentum tensor are: Let us first consider V V V .By using the notation for the currents J (s,q) , J(s,q) , this corresponds to the general three-point function The general ansatz for this correlation function, according to (2.34) is Using the formalism outlined in subsection 2.2.2, all information about this correlation function is encoded in the following polynomial: Using Mathematica we solve (2.57) for the chosen spin representations of the currents and substitute each solution into the generating function (2.56).This provides us with the following list of (linearly dependent) polynomial structures: Next, we systematically apply the linear dependence relations (2.50) to these lists, reducing them to the following sets of linearly independent structures: Note that application of the linear-dependence relations eliminates all terms involving Z i in this case.Since this correlation function is composed of fields in the (s, s) representation, the solutions for the three-point function may be split up into parity-even and parity-odd contributions.To do this we construct linear combinations for the polynomial Ĥ(X; U, V, W ) which are even/odd under inversion in accordance with (2.79): We note here (and in all other examples) that the parity-even contributions possess the complex coefficients A i , while the parity-odd solutions possess the complex coefficients B i .
It can be explicitly checked that these structures possess the appropriate transformation properties.Next, since the correlation function is overall real, we must impose the reality condition (2.74).As a result, we find that the parity-even coefficients A i are purely real, i.e., A i = a i , while the parity-odd coefficients B i are purely imaginary, i.e., B i = ib i .
We must now impose the conservation of the currents.Following the procedure outlined in 2.2.2 we obtain a linear system in the coefficients a i , b i which can be easily solved computationally.We find the following solution for H(X; U, V, W ) consistent with conservation on all three points: The only remaining constraints to impose are symmetries under permutations of spacetime points, which apply when the currents in the three-point function are identical, i.e. when J = J = J .After imposing (2.72), (2.73),only the structure corresponding to the coefficient b 1 survives.However, the a 1 , a 2 structures can exist if the currents are nonabelian.This is consistent with the results of [1,2,13]. 4he next example to consider is the mixed correlator V V T .To study this case we may examine the correlation function J (1,0) J (1,0) J (2,0) .

Correlation function J
Using the general formula, the ansatz for this three-point function is: (3.9)All information about this correlation function is encoded in the following polynomial: 2) . (3.10) After solving (2.57), we find the following linearly dependent polynomial structures: We now systematically apply the linear dependence relations (2.50)-(2.55) to obtain the linearly independent structures (3.12) Next, we construct the following parity-even and parity-odd linear combinations which comprise the polynomial Ĥ(X; U, V, W ): We now impose conservation on all three points to obtain the final solution for H(X; U, V, W ) In this case, only the parity-even structures (proportional to a 1 and a 2 ) survive after setting J = J .Hence, this correlation function is fixed up to two independent parityeven structures with real coefficients.
The number of polynomial structures increases rapidly for increasing s i , and for the three-point functions T T V , T T T we will present only the linearly independent structures and the final results after imposing parity, reality, and conservation on all three points.For T T V we may consider the correlation function J (2,0) J (2,0) J (1,0) , which is constructed from the following list of linearly independent structures: We now construct linearly independent parity-even and parity-odd solutions consistent with (2.79).Then, after imposing all the constraints due to reality and conservation, we obtain the final solution for H(X; U, V, W ): After setting J = J and imposing the required symmetries under the exchange of x 1 and x 2 we find that b 1 = 0, while a 1 , a 2 remain unconstrained.Hence, the correlation function T T V is fixed up to two parity-even structures with real coefficients.
The final fundamental three-point function to study is T T T , and for this we analyse the correlation function J (2,0) J (2,0) J (2,0) .In this case there are 15 linearly independent structures to consider: three points we obtain the following solution for H(X; U, V, W ): In this case only three of the structures (corresponding to the real coefficients a 1 , a 2 , a 3 ) survive the point-switch symmetries upon exchange of x 1 , x 2 and x 3 .Hence, T T T is fixed up to three parity-even structures with real coefficients.
In all cases we note that the number of independent structures (prior to imposing exchange symmetries) is 2 min(s 1 , s 2 , s 3 )+1 in general, where min(s 1 , s 2 , s 3 )+1 are parityeven and min(s 1 , s 2 , s 3 ) are parity-odd.These results are in agreement with [1,[13][14][15]18] in terms of the number of independent structures, however, our construction of the three-point function is quite different.

Spin-3/2 current correlators
In this section we will evaluate three-point functions involving conserved fermionic currents.The most important examples of fermionic conserved currents in 4D CFT are the supersymmetry currents, Q m,α , Qm, α, which appear in N -extended superconformal field theories.Such fields are primary with dimension ∆ Q = ∆ Q = 7/2, and satisfy the conservation equations ∂ m Q m,α = 0, ∂ m Qm, α = 0.In spinor notation, we have: The correlation functions involving supersymmetry currents, vector currents, and the energy-momentum tensor are of fundamental importance.The four possible three-point functions involving Q, V and T which are of interest in N = 1 superconformal field theories are: These three-point functions were analysed in [17] using a similar approach, but we present them again here for completeness and to demonstrate our general formalism.Note that in the subsequent analysis we assume only conformal symmetry, not supersymmetry.
We now present an explicit analysis of the general structure of correlation functions involving Q, Q, V and T that are compatible with the constraints of conformal symmetry and conservation equations.Using our conventions for the currents, we recall that Q ≡ J (3/2,1) , Q ≡ J(3/2,1) .Let us first consider QQV , for which we may analyse the general structure of the correlation function Using the general formula, the ansatz for this three-point function: Using the formalism outlined in 2.2.2, all information about this correlation function is encoded in the following polynomial: After solving (2.57), we find the following linearly dependent polynomial structures in the even and odd sectors respectively: Next we systematically apply the linear dependence relations (2.50)-(2.55)and obtain the following linearly independent structures: We now impose conservation on all three points and find that the solution for H(X; U, V, W ) is unique up to a complex coefficient, A 1 = a 1 + iã 1 : Q 3 P2 Q1 Q3 . (3.25) However, this three-point function is not compatible with the point-switch symmetry associated with setting J = J .Therefore we conclude that the three-point function QQV must vanish in general.
Correlation function J (3/2,1) J (3/2,1) J (1,0) : Using the general formula we obtain the following ansatz: The tensor three-point function is encoded in the following polynomial: After solving (2.57), we find the following linearly dependent polynomial structures: (3.28) Next we systematically apply the linear dependence relations (2.50) to this list, which results in the following linearly independent structures: We now construct the ansatz for this three-point function using the linearly independent structures above.After imposing conservation on all three points the final solution is A 1 (3.30) Therefore we see that the correlation function J (3/2,1) J (3/2,1) J (1,0) and, hence, Q QV , is fixed up to three independent complex coefficients.After imposing the combined pointswitch/reality condition on Q and Q, we find that the complex coefficients A i must be purely imaginary, i.e., A i = iã i .Hence, the correlation function Q QV is fixed up to three independent real parameters.
Next we determine the general structure of QQT and Q QT , which are associated with the correlation functions J (3/2,1) J (3/2,1) J (2,0) , J (3/2,1) J (3/2,1) J (2,0) respectively using our general formalism.Since the number of structures grows rapidly with spin, we will simply present the final results after conservation.For J (3/2,1) J (3/2,1) J (2,0) we obtain a single independent structure (up to a complex coefficient): This solution is manifestly compatible with the point-switch symmetry resulting from setting J = J , hence, QQT is unique up to a complex parameter.On the other hand, for J (3/2,1) J (3/2,1) J (2,0) we obtain four independent conserved structures proportional to complex coefficients (3.32) After imposing the combined point-switch/reality condition, we find that the complex coefficients A i must be purely real.Hence, the three-point function Q QT is fixed up to four independent real parameters.The results (3.25), (3.30), (3.31), (3.32) are in agreement with those found in [17].
3.2 General structure of three-point functions for arbitrary spins In four dimensions, three-point correlation functions of bosonic higher-spin conserved currents have been analysed in the following publications [13,18] (see [20,24,34] for supersymmetric results).For three-point functions involving bosonic currents J (s,0) = J α(s) α(s) , the general structure of the three-point function J (s 1 ,0) J (s 2 ,0) J (s 3 ,0) was found to be fixed up to the following form [18,37,38]: where a I are real coefficients and J (s 1 ,0) J (s 2 ,0) J (s 3 ,0) I are linearly independent conserved structures.5Among these 2 min(s 1 , s 2 , s 3 )+1 structures, min(s 1 , s 2 , s 3 )+1 are parity-even while min(s 1 , s 2 , s 3 ) are parity odd.For correlation functions involving identical fields we must also impose point-switch symmetries.The following classification holds: • For three-point functions J (s,0) J (s,0) J (s,0) there are 2s+1 conserved structures, s+1 being parity even and s being parity odd.When the fields coincide, i.e.J = J the number of structures is reduced to the s + 1 parity-even structures in the case when the spin s is even, or to the s parity-odd structures in the case when s is odd.
• For three-point functions J (s 1 ,0) J (s 1 ,0) J (s 2 ,0) , there are 2 min(s 1 , s 2 ) + 1 conserved structures, min(s 1 , s 2 ) + 1 being parity even and min(s 1 , s 2 ) being parity odd.For J = J , the number of structures is reduced to the min(s 1 , s 2 ) + 1 parity-even structures in the case when the spin s 2 is even, or to the min(s 1 , s 2 ) parity-odd structures in the case when s 2 is odd.
Note that the above classification is consistent with the results of [13], and we have explicitly reproduced them up to s i = 10 in our computational approach.Now let us discuss three-point functions involving currents with q = 1, which define "supersymmetry-like" fermionic higher-spin currents.The possible correlation functions that we can construct from these are J (s 1 ,1) J (s 2 ,1) J (s 3 ,0) and J (s 1 ,1) J (s 2 ,1) J (s 3 ,0) .Note that for s 1 = s 2 = 3/2 and s 3 = 1, 2 we obtain the familiar three-point functions (3.20).Based on our computational analysis we found that the three-point function J (s 1 ,1) J (s 2 ,1) J (s 3 ,0) is fixed up to a unique structure after conservation in general.On the other hand, we found that three-point functions of the form J (s 1 ,1) J (s 2 ,1) J (s 3 ,0) are fixed up to 2 min(s 1 , s 2 , s 3 ) + 1 independent conserved structures.It's important to note that for these three-point functions there is no notion of parity-even/odd structures.
We now dedicate the remainder of this section to classifying the number of independent structures in the general three-point functions J (s 1 ,q 1 ) J (s 2 ,q 2 ) J (s 3 ,q 3 ) , J (s 1 ,q 1 ) J (s 2 ,q 2 ) J (s 3 ,q 3 ) , for arbitrary (s i , q i ).We investigated the general structure of these three-point functions up to s i = 10.Provided that the inequalities (2.61) are satisfied, we conjecture that the following classification holds in general: • For three-point functions J (s 1 ,q 1 ) J (s 2 ,q 2 ) J (s 3 ,q 3 ) , J (s 1 ,q 1 ) J (s 2 ,q 2 ) J (s 3 ,q 3 ) with q 1 = q 2 = q 3 , there is a unique solution in general.Similarly, the three-point function is also unique for the cases: i) q 1 = 0, q 2 = q 3 , and ii) q 1 = q 2 = 0 with q 3 = 0.
• For three-point functions J (s 1 ,q) J (s 2 ,q) J (s 3 ,0) there is a unique solution up to a complex coefficient.However, for the case where s 1 = s 2 (fermionic or bosonic) and J = J , the structure survives the resulting point-switch symmetry only when s 3 is an even integer.
• For three-point functions J (s 1 ,q) J (s 2 ,q) J (s 3 ,0) we obtain quite a non-trivial result which we will now explain.The number of structures, N (s 1 , s 2 , s 3 ; q), obeys the following formula: where s 1 , s 2 are simultaneously integer/half-integer, for integer s 3 .This formula can be arrived at using the following method.Let us fix s 1 , s 2 and let q ≥ 2. By varying s 3 and computing the resulting conserved three-point function, one can notice that if s 3 lies within the interval then the number of structures is decreased from 2 min(s 1 , s 2 , s 3 ) + 1 by For s 3 outside the interval (3.36) there is always 2 min(s 1 , s 2 , s 3 ) + 1 structures in general.It should also be noted that (3.35) is also valid for q = 0, 1 (by virtue of the max() function).In these cases the additional term does not contribute and we obtain N (s 1 , s 2 , s 3 ; 0) = N (s 1 , s 2 , s 3 ; 1) = 2 min(s 1 , s 2 , s 3 ) + 1.
As examples, below we tabulate the number of structures in the conserved threepoint functions J (s 1 ,q) J (s 2 ,q) J (s 3 ,0) for some fixed s 1 , s 2 while varying q and s 3 .Let us recall that q is necessarily even/odd when s is integer/half-integer valued.In addition, since J (s,q) := J α(s+ q 2 ) α(s− q 2 ) it follows that the maximal allowed value of q in the above correlation function is 2 min(s 1 , s 2 )−2.Explicit solutions for particular cases are presented in appendix B.
Table 2: No. of structures in J (s 1 ,q) J (s 2 ,q) J (s 3 ,0) for s 1 = 9/2, s 2 = 11/2.• δN = 1 The highlighted values are within the interval (3.36) defined by s 1 , s 2 and q, and we have used colour to identify the pattern in the number of structures.Analogous tables can be constructed for any choice of s 1 , s 2 and it is easy to see that the results are consistent with the general formula (3.35), which appears to hold for all such correlators within the bounds of our computational limitations (s i ≤ 10).
• For three-point functions J (s 1 ,q) J (s 1 ,q) J (s 2 ,0) the number of structures adheres to the formula (3.35).However, for J = J , we must impose the combined pointswitch/reality condition.After imposing this constraint we find that the free complex parameters must be purely real/imaginary for s 2 even/odd.
The above classification appears to be complete, and we have not found any other permutations of fields/spins which give rise to new results.

Correlation function O O J (s,0) :
Let O, O be scalar operators of dimension ∆ 1 and ∆ 2 respectively.We consider the three-point function O O J (s,0) .According to the formula (2.61), a three-point function can be constructed only if J is in the (s, s) representation.Using the general formula, the ansatz for this three-point function is: O(x 1 ) O (x 2 ) J γ(s) γ(s) (x 3 ) = 1 (x 2 13 ) ∆ 1 (x 2 23 ) ∆ 2 H γ(s) γ(s) (X 12 ) .We recall that H satisfies the homogeneity property H(X) = X s+2−∆ 1 −∆ 2 Ĥ(X), where Ĥ(X) is homogeneous degree 0. The only possible structure for Ĥ(X) is: where A is a complex coefficient.After imposing conservation on x 3 using the methods outlined in subsection 2.2.2, we find Hence, we find that this three-point function is compatible with conservation on x 3 only for ∆ 1 = ∆ 2 .When the scalars O, O coincide, then the solution satisfies the pointswitch symmetry associated with exchanging x 1 and x 2 only for even s.This result is in agreement with [16].
Correlation function ψ ψ J (s,q) : Let ψ, ψ be spinor operators of dimension ∆ 1 and ∆ 2 respectively.We now consider the three-point function ψ ψ J (s,q) .According to the formula (2.61), a three-point function can be constructed only if J belongs to the representations (s, s), (s−1, s+1) or (s+1, s−1) (the latter two corresponding to q = 2).First consider the (s, s) representation.Using the general formula, the ansatz for this three-point function is: We recall that H satisfies the homogeneity property H(X) = X s+2−∆ 1 −∆ 2 Ĥ(X), where Ĥ(X) is homogeneous degree 0. In this case there are two possible linearly independent structures for Ĥ(X): Ĥ(X; U, V, W ) = A 1 P 2 P1 Z s−1 where A 1 and A 2 are complex coefficients.After imposing conservation on x 3 using the methods outlined in subsection 2.2.2, we find (C.8) Hence, we find that this three-point function is automatically compatible with conservation on x 3 for ∆ 1 = ∆ 2 .For ∆ 1 = ∆ 2 it is simple to see that conservation is satisfied only for s = 1, which results in A 1 = −A 2 and, hence, the solution is unique.However, for s > 1 there is no solution in general.In the case where ψ = ψ , we also have to impose the combined point-switch/reality condition, which results in the coefficients A i being purely real/imaginary for s even/odd.This result is consistent with [16].Now let us consider the (s+1, s−1) representation, with s > 1.Note that the analysis for (s − 1, s + 1) is essentially identical and will be omitted.Using the general formula, the ansatz for this three-point function is: (C.9) All information about this correlation function is encoded in the following polynomial: H(X; U, V, W ) = H αβγ(s+1) γ(s−1) (X) U αV β W γ(s+1) γ(s−1) .(C.10) We recall that H satisfies the homogeneity property H(X) = X s+2−∆ 1 −∆ 2 Ĥ(X), where Ĥ(X) is homogeneous degree 0. In this case there is only one possible structure for Ĥ(X): where A is a complex coefficient.After imposing conservation on x 3 using the methods outlined in subsection 2.2.2, we find Hence, we find that this three-point function is automatically compatible with conservation on x 3 for ∆ 2 = ∆ 1 − 1.For ∆ 2 = ∆ 1 − 1 there is no solution in general (recall that s > 1).This result is also consistent with [16].
.73) (iv) Reality condition:If the fields in the correlation function belong to the (s, s) representation, then the three-point function must satisfy the reality condition H(X; U, V, W ) = H(X; U, V, W ) .