Deconstructing Conformal Blocks in 4D CFT

We show how conformal partial waves (or conformal blocks) of spinor/tensor correlators can be related to each other by means of differential operators in four dimensional conformal field theories. We explicitly construct such differential operators for all possible conformal partial waves associated to four-point functions of arbitrary traceless symmetric operators. Our method allows any conformal partial wave to be extracted from a few"seed"correlators, simplifying dramatically the computation needed to bootstrap tensor correlators.


Introduction
There has been a revival of interest in recent years in four dimensional (4D) Conformal Field Theories (CFTs), after the seminal paper [1] resurrected the old idea of the bootstrap program [2,3]. A 4D CFT is determined in terms of its spectrum of primary operators and the coefficients entering three-point functions among such primaries. Once this set of CFT data is given, any correlator is in principle calculable. Starting from this observation, ref. [1] has shown how imposing crossing symmetry in four point functions can lead to non-trivial sets of constraints on the CFT data. These are based on first principles and apply to any CFT, with or without a Lagrangian description. Although any correlator can in principle be "bootstrapped", in practice one has to be able to sum, for each primary operator exchanged in the correlator in some kinematical channel, the contribution of its infinite series of descendants. Such contribution is often called a conformal block. In fact, the crucial technical ingredient in ref. [1] was the work of refs. [4,5], where such conformal blocks have been explicitly computed for scalar four-point functions. Quite remarkably, the authors of refs. [4,5] were able to pack the contributions of traceless symmetric operators of any spin into a very simple formula.
Since ref. [1], there have been many developments, both analytical [6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25] and numerical [26][27][28][29][30][31][32][33][34][35][36][37][38][39][40][41] in the 4D bootstrap. All numerical studies are still based on identical scalar correlators, unless supersymmetry or global symmetries are present. 1 There is an obvious reason for this limitation. Determining the conformal blocks relevant for four-point functions involving tensor primary operators is significantly more complicated. First of all, contrary to their scalar counterpart, tensor four-point correlators are determined in terms of several functions, one for each independent allowed tensor structure. Their number N 4 grows very rapidly with the spin of the external operators. The whole contribution of primary operators in any given channel is no longer parametrized by a single conformal block as in the scalar case, but in general by N 4 × N 4 conformal blocks, N 4 for each independent tensor structure. For each exchanged primary operator, it is convenient not to talk of individual conformal blocks but of Conformal Partial Waves (CPW), namely the entire contribution given by several conformal blocks, one for each tensor structure. Second, the exchanged operator is no longer necessarily traceless symmetric, but can be in an arbitrary representation of the 4D Lorentz group, depending on the external operators and on the channel considered.
CPW can be determined in terms of the product of two three-point functions, each involving two external operators and the exchanged one. If it is possible to relate a three-point function to another simpler one, a relation between CPW associated to different four-point functions can be obtained. Using this simple observation, building on previous work [8], in ref. [9] the CPW associated to a correlator of traceless symmetric operators (in arbitrary space-time dimensions), which exchange a traceless symmetric operator, have been related to the scalar conformal block of refs. [4,5]. Despite this significant progress, bootstrapping tensor four-point functions in 4D requires the knowledge of the CPW associated to the exchange of non-traceless symmetric operators. Even for traceless symmetric exchange, the methods of refs. [8,9] do not allow to study correlators with external non-traceless symmetric fields (although generalizations that might do that have been proposed, see ref. [19]).
The aim of this paper is to make a step forward and generalize the relation between CPW found for traceless symmetric operators in ref. [9] to arbitrary CPW in 4D CFTs. We will perform this task by using the 6D embedding formalism in terms of twistors. Our starting point is the recent general classification of 3-point functions found in ref. [21]. We will see how three-point functions of spinors/tensors can be related to three-point functions of lower spin fields by means of differential operators. We explicitly construct a basis of differential operators that allows one to express any three-point function of two traceless symmetric and an arbitrary bosonic operator O l,l with l =l, in terms of "seed" three-point functions, that admit a unique tensor structure. This would allow to express all the CPW entering a four-point function of traceless symmetric correlators in terms of a few CPW seeds. We do not attempt to compute such seeds explicitly, although it might be done by developing the methods of refs. [4,5].
The structure of the paper is as follows. In section 2 we will briefly review the 6D embedding formalism in twistor space in index-free notation and the results of ref. [21] on the three-point function classification. In section 3 we recall how a relation between three-point functions leads to a relation between CPW. We introduce our differential operators in section 4. We construct an explicit basis of differential operators in section 5 for external symmetric traceless operators. In subsection 5.1 we reproduce (and somewhat improve) the results of ref. [9] in our formalism where the exchanged operator is traceless symmetric and then pass to the more involved case of mixed tensor exchange in subsection 5.2. In section 6 we discuss the basis of the tensor structures of four-point functions and propose a set of seed CPW needed to get CPW associated with the exchange of a bosonic operator O l,l . A couple of examples are proposed in section 7. In subsection 7.1 we consider a four fermion correlator and in subsection 7.2 we schematically deconstruct spin one and spin two correlators, and show how to impose their conservation. We conclude in section 8, where we discuss in particular the computations yet to be done to bootstrap tensor correlators in 4D CFTs. A (non-exhaustive) list of relations between SU (2, 2) invariants entering four-point functions is listed in appendix A.
A 4D primary operator Oβ 1 ...βl α 1 ...α l with scaling dimension ∆ in the (l,l) representation of the Lorentz group can be embedded in a 6D multi-twistor field O a 1 ...a l b 1 ...bl , homogeneous of degree n = ∆ + (l +l)/2, as follows: In eq.(2.1), 6D and 4D coordinates are denoted as X M and x µ , where x µ = X µ /X + , X and X are 6D twistor space-coordinates defined as in terms of the 6D chiral Gamma matrices Σ M and Σ M (see Appendix A of ref. [21] for further details). One has XX = XX = X M X M = X 2 , which vanishes on the null 6D cone.
It is very useful to use an index-free notation by defining The 4D three-point functions are conveniently encoded in their scalar 6D counterpart O 1 O 2 O 3 which must be a sum of SU (2, 2) invariant quantities constructed out of the X i , S i andS i , with the correct homogeneity properties under rescaling. Notice that quantities proportional toS i X i , X i S i orS i S i (i = 1, 2, 3) are projected to zero in 4D. The non-trivial SU (2, 2) possible invariants are (i = j = k, indices not summed) [11]: Two-point functions are easily determined. One has where X ij ≡ X i · X j and τ i ≡ ∆ i + (l i +l i )/2. As can be seen from eq.(2.10), any operator O l,l has a non-vanishing two-point function with a conjugate operator Ol ,l only. The main result of ref. [21] can be recast in the following way. The most general three-point where In eq.(2.12), K 3 is a kinematic factor that depends on the scaling dimension and spin of the external fields, 12 12 X a 13 13 X a 23

23
, The points X1, X2 and X3 are assumed to be distinct.
with a ij = (τ i + τ j − τ k )/2, i = j = k. The index s runs over all the independent tensor structures parametrized by the integers m ij and n i , each multiplied by a constant OPE coefficient λ s . The invariants C i,jk equal to one of the three-index invariants (2.6)-(2.8), depending on the value of of the external fields. Three-point functions are non-vanishing only when ∆l is an even integer [62,21]. We have A redundance is present for ∆l = 0. It can be fixed by demanding, for instance, that one of the three integers n i in eq.(2.12) vanishes. The total number of K i,jk 's (K i,jk 's) present in the correlator for ∆l > 0 (∆l < 0) equal ∆l/2 (−∆l/2). The number of tensor structures is given by all the possible allowed choices of nonnegative integers m ij and n i in eq.(2.11) subject to the above constraints and the ones coming from matching the correct powers of S i andS i for each field. The latter requirement gives in total six constraints. Conserved 4D operators are encoded in multitwistors O that satisfy the current conservation condition When eq.(2.15) is imposed on eq.(2.11), we generally get a set of linear relations between the OPE coefficients λ s 's, which restrict the possible allowed tensor structures in the three point function. Under a 4D parity transformation, the invariants (2.5)-(2.8) transform as follows: (2.16)

Relation between CPW
A CFT is defined in terms of the spectrum of primary operators, their scaling dimensions ∆ i and SL(2, C) representations (l i ,l i ), and OPE coefficients, namely the coefficients entering the three-point functions among such primaries. Once this set of CFT data is given, any correlator is in principle calculable. Let us consider for instance the 4-point function of four primary tensor operators: In eq.(3.1) we have schematically denoted by I i the Lorentz indices of the operators O i (x i ), is a kinematical factor, u and v are the usual conformally invariant cross ratios

3)
T I 1 I 2 I 3 I 4 n (x i ) are tensor structures and τ i are defined below eq.(2.10). These are functions of the x i 's and can be kinematically determined. Their total number N 4 depends on the Lorentz properties of the external primaries. For correlators involving scalars only, one has N 4 = 1, but in general N 4 > 1 and rapidly grows with the spin of the external fields. For instance, for four traceless symmetric operators with identical spin l, one has N 4 (l) ∼ l 7 for large l [21]. All the non-trivial dynamical information of the 4-point function is encoded in the N 4 functions g n (u, v). In any given channel, by using the OPE we can write the 4-point function (3.1) in terms of the operators exchanged in that channel. In the s-channel , for instance, we have where p and q run over the possible independent tensor structures associated to the three point functions O 1 O 2 O r and Ōr O 3 O 4 , whose total number is N 12 3r and N 34 3r respectively, 3 the λ's being their corresponding structure constants, and r and O r runs over the number of primary operators that can be exchanged in the correlator. We divide the (infinite) sum over the exchanged operators in a finite sum over the different classes of representations that can appear, e.g. (l, l), (l + 2, l), etc., while the sum over O r includes the sum over the scaling dimension and spin l of the operator exchanged within the class r. For example, four-scalar correlators can only exchange traceless symmetric operators and hence the sum over r is trivial. Finally, in eq.(3.4) W (p,q) are the so-called CPW associated to the four-point function. They depend on the external as well as the exchanged operator scaling dimension and spin, dependence we omitted in order not to clutter further the notation. 4 By comparing eqs.(3.1) and (3.4) one can infer that the number of allowed tensor structures in three and four-point functions is related: 5 3 Strictly speaking these numbers depend also on Or, particularly on its spin. When the latter is large enough, however, N 12 3r and N 34 3r are only functions of the external operators. 4 For further simplicity, in what follows we will often omit the subscript indicating the external operators associated to the CPW. 5 We do not have a formal proof of eq.(3.5), although the agreement found in ref. [21] using eq.(3.5) in different channels is a strong indication that it should be correct.
There are several CPW for each exchanged primary operator O r , depending on the number of allowed 3-point function structures. They encode the contribution of all the descendant operators associated to the primary O r . Contrary to the functions g n (u, v) in eq.(3.1), the CPW do not carry dynamical information, being determined by conformal symmetry alone. They admit a parametrization like the 4-point function itself, where G (p,q) Or,n (u, v) are conformal blocks depending on u and v and on the dimensions and spins of the external and exchanged operators. Once the CPW are determined, by comparing eqs.(3.1) and (3.4) we can express g n (u, v) in terms of the OPE coefficients of the exchanged operators. This procedure can be done in other channels as well, (13 − 24) and (14 − 23). Imposing crossing symmetry by requiring the equality of different channels is the essence of the bootstrap approach.
The computation of CPW of tensor correlators is possible, but technically is not easy. In particular it is desirable to have a relation between different CPW, so that it is enough to compute a small subset of them, which determines all the others. In order to understand how this reduction process works, it is very useful to embed the CPW in the 6D embedding space with an index-free notation. We use here the formalism in terms of twistors as reviewed in section 2. It is useful to consider the parametrization of CPW in the shadow formalism [63][64][65][66]. It has been shown in ref. [11] that a generic CPW can be written in 6D as (3.7) In eq.(3.7), O i (X i ) = O i (X i , S i ,S i ) are the index-free 6D fields associated to the 4D fields O i (x i ), O r (X, S,S) andŌr(Y, T,T ) are the exchanged operator and its conjugate, G is a sort of "propagator", function of X, Y and of the twistor derivatives ∂/∂S, ∂/∂T , ∂/∂S and ∂/∂T , and the subscripts p and q label the three-point function tensor structures. Finally, in order to remove unwanted contributions, the transformation X 12 → e 4πi X 12 should be performed and the integral should be projected to the suitable eigenvector under the above monodromy. We do not provide additional details, which can be found in ref. [11], since they are irrelevant for our considerations. Suppose one is able to find a relation between three-point functions of this form: where D pp ′ is some operator that depends on X 12 , S 1,2 ,S 1,2 and their derivatives, but is crucially independent of X, S, andS, and O ′ i (X i ) are some other, possibly simpler, tensor operators. As long as the operator D pp ′ (X 12 , S 1,2 ,S 1,2 ) does not change the monodromy properties of the integral, one can use eq.(3.8) in both three-point functions entering eq.(3.7) and move the operator D pp ′ outside the integral. In this way we get, with obvious notation, Using the embedding formalism in vector notation, ref. [9] has shown how to reduce, in any space-time dimension, CPW associated to a correlator of traceless symmetric operators which exchange a traceless symmetric operator to the known CPW of scalar correlators [4,5]. Focusing on 4D CFTs and using the embedding formalism in twistor space, we will see how the reduction of CPW can be generalized for arbitrary external and exchanged operators.

Differential Representation of Three-Point Functions
We look for an explicit expression of the operator D pp ′ defined in eq.(3.8) as a linear combination of products of simpler operators. They must raise (or more generically change) the degree in S 1,2 and have to respect the gauge redundancy we have in the choice of O. As we recalled in subsection 2, multitwistors O andÔ of the form where G and G ′ are some other multi-twistors fields, are equivalent uplifts of the same 4D tensor field. Eq.(3.8) is gauge invariant with respect to the equivalence classes (4.1) only if we demand It is useful to classify the building block operators according to their value of ∆l, as defined in eq.(2.14).
At zero order in derivatives, we have three possible operators, with ∆l = 0: At first order in derivatives (in X and S), four operators are possible with ∆l = 0: The extra two terms in the last two lines of eq.(4.4) are needed to satisfy the condition (4.2). The SU (2, 2) symmetry forbids any operator at first order in derivatives with ∆l = ±1. When ∆l = 2, we have the two operators and their conjugates with ∆l = −2: The operator √ X 12 just decreases the dimensions at both points 1 and 2 by one half. The operator I 12 increases by one the spinl 1 and by one l 2 . The operator D 1 increases by one the spin l 1 and by onel 1 , increases by one the dimension at point 1 and decreases by one the dimension at point 2. The operator D 1 increases by one the spin l 1 and by one the spinl 1 and it does not change the dimension of both points 1 and 2. The operator d 1 increases by one the spin l 2 and decreases by onel 1 , decreases by one the dimension at point 1 and does not change the dimension at point 2. The action of the remaining operators is trivially obtained by 1 ↔ 2 exchange or by conjugation.
Two more operators with ∆l = 2 are possible: together with their conjugates with ∆l = −2. We will shortly see that the operators (4.7) are redundant and can be neglected.
The above operators satisfy the commutation relations Some other commutators are trivially obtained by exchanging 1 and 2 and by the parity transformation (4.14). The operators √ X 12 , I 12 and I 21 commute with all the differential operators. Acting on the whole correlator, we have and hence the above differential operators, together with X 12 and I 12 I 21 , form a closed algebra when acting on three-point correlators. Useful information on conformal blocks can already be obtained by considering the rather trivial operator √ X 12 . For any three point function tensor structure, we have where a is an integer (in order not to induce a monodromy for X 12 → e 4πi X 12 ) and the super- Using eqs.(4.10) and (3.9), we get for any 4D CPW and pair of integers a and b: In terms of the conformal blocks defined in eq.(3.6) one has where the superscripts indicate the shifts in dimension in the four external operators. Equation Or,n on the external operator dimensions ∆ i . The conformal blocks can be periodic functions of ∆ 1 , ∆ 2 and ∆ 3 , ∆ 4 , but can arbitrarily depend on ∆ 1 − ∆ 2 , ∆ 3 − ∆ 4 . This is in agreement with the known form of scalar conformal blocks. Since in this paper we are mostly concerned in deconstructing tensor structures, we will neglect in the following the operator √ X 12 . The set of differential operators is redundant, namely there is generally more than 1 combination of products of operators that lead from one three-point function structure to another one. In particular, without any loss of generality we can forget about the operators (4.7), since their action is equivalent to commutators of d i and D j . On the other hand, it is not difficult to argue that the above operators do not allow to connect any three-point function structure to any other one. For instance, it is straightforward to verify that there is no way to connect a three-point correlator with one (l,l) field to another correlator with a (l ± 1,l ∓ 1) field, with the other fields left unchanged. This is not an academic observation because, as we will see, connections of this kind will turn out to be useful in order to simplify the structure of the CPW seeds. The problem is solved by adding to the above list of operators the following second-order operator with ∆l = 0: and its conjugate ∇ 21 . The above operators transform as follows under 4D parity: It is clear that all the operators above are invariant under the monodromy X 12 → e 4πi X 12 .
The addition of ∇ 12 and ∇ 21 makes the operator basis even more redundant. It is clear that the paths connecting two different three-point correlators that make use of the least number of these operators are preferred, in particular those that also avoid (if possible) the action of the second order operators ∇ 12 and ∇ 21 . We will not attempt here to explicitly construct a minimal differential basis connecting two arbitrary three-point correlators. Such an analysis is in general complicated and perhaps not really necessary, since in most applications we are interested in CPW involving external fields with spin up to two. Given their particular relevance, we will instead focus in the next section on three-point correlators of two traceless symmetric operators with an arbitrary field O (l,l) .

Differential Basis for Traceless Symmetric Operators
In this section we show how three-point correlators of two traceless symmetric operators with an arbitrary field O (l 3 ,l 3 ) can be reduced to seed correlators, with one tensor structure only. We first consider the case l 3 =l 3 , and then go on with l 3 =l 3 .

Traceless Symmetric Exchanged Operators
The reduction of traceless symmetric correlators to lower spin traceless symmetric correlators has been successfully addressed in ref. [9]. In this subsection we essentially reformulate the results of ref. [9] in our formalism. This will turn out to be crucial to address the more complicated case of mixed symmetry operator exchange. Whenever possible, we will use a notation as close as possible to that of ref. [9], in order to make any comparison more transparent to the reader. Three-point correlators of traceless symmetric operators can be expressed only in terms of the SU (2, 2) invariants I ij and J i,jk defined in eqs.(2.5)-(2.8), since ∆l defined in eq.(2.14) vanishes. It is useful to consider separately parity even and parity odd tensor structures. Given the action of parity, eq.(2.16), the most general parity even tensor structure is given by products of the following invariants: In ref. [21], eq.(5.2) has been used to define an independent basis where no tensor structure contains the three SU (2, 2) invariants J 1,23 , J 2,31 and J 3,12 at the same time. A more symmetric and convenient basis is obtained by using eq.(5.2) to get rid of the first factor in eq.(5.1). We define the most general parity even tensor structure of traceless symmetric tensor correlator as    where l i and ∆ i are the spins and scaling dimensions of the fields, the kinematical factor K 3 is defined in eq.(2.13) and Notice the similarity of eq.(5.3) with eq.(3.15) of ref. [9], with (I ij I ji ) → H ij and J i,jk → V i,jk . The structures (5.3) can be related to a seed scalar-scalar-tensor correlator. Schematically    where D is a sum of products of the operators introduced in section 4. Since symmetric traceless correlators have ∆l = 0, it is natural to expect that only the operators with ∆l = 0 defined in eqs.(4.3) and (4.4) will enter in D. Starting from the seed, we now show how one can iteratively construct all tensor structures by means of recursion relations. The analysis will be very similar to the one presented in ref. [9] in vector notation. We first construct tensor structures with m 13 = m 32 = 0 for any l 1 and l 2 by iteratively using the relation (analogue of eq.(3.27) in ref. [9], with D 1 → D 12 and D 1 → D 11 ) The analogous equation with D 2 and D 2 is obtained from eq.(5.6) by exchanging 1 ↔ 2 and changing sign of the coefficients in the right hand side of the equation. The sign change arises from the fact that J 1,23 → −J 2,31 , J 2,31 → −J 1,23 and J 3,12 → −J 3,12 under 1 ↔ 2. Hence structures that differ by one spin get a sign change. This observation applies also to eq.(5.8) below. Structures with m 12 > 0 are deduced using (analogue of eq(3.28) in ref. [9])    Structures with non-vanishing m 13 (m 23 ) are obtained by acting with the operator D 1 (D 2 ): and is the analogue of eq (3.29) in ref. [9]. In this way all parity even tensor structures can be constructed starting from the seed correlator.
Let us now turn to parity odd structures. The most general parity odd structure is given by    Since the parity odd combination (I 12 I 23 I 31 + I 21 I 32 I 13 ) commutes with D 1,2 and D 1,2 , the recursion relations found for parity even structures straightforwardly apply to the parity odd ones. One could define a "parity odd seed" and from here construct all the parity odd structures. Notice that the parity odd seed cannot be obtained by applying only combinations of D 1,2 , D 1,2 and (I 12 I 21 ), because these operators are all invariant under parity, see eq.(4.14). This explains the appearance of the operators d i andd i in eq.(5.10). The counting of parity even and odd structures manifestly agrees with that performed in ref. [8].
Once proved that all tensor structures can be reached by acting with operators on the seed correlator, one might define a differential basis which is essentially identical to that defined in eq.(3.31) of ref. [9]: where ∆ ′ 1 = ∆ 1 + l 1 + m 23 − m 13 , ∆ ′ 2 = ∆ 2 + l 2 + m 13 − m 23 . The recursion relations found above have shown that the differential basis (5.11) is complete: all parity even tensor structures can be written as linear combinations of eq.(5.11). The dimensionality of the differential basis matches the one of the ordinary basis for any spin l 1 , l 2 and l 3 . Since both bases are complete, the transformation matrix relating them is ensured to have maximal rank. Its determinant, however, is a function of the scaling dimensions ∆ i and the spins l i of the fields and one should check that it does not vanish for some specific values of ∆ i and l i . We have explicitly checked up to l 1 = l 2 = 2 that for l 3 ≥ l 1 + l 2 the rank of the transformation matrix depends only on ∆ 3 and l 3 and never vanishes, for any value of ∆ 3 allowed by the unitarity bound [67]. On the other hand, a problem can arise when l 3 < l 1 + l 2 , because in this case a dependence on the values of ∆ 1 and ∆ 2 arises and the determinant vanishes for specific values (depending on the l i 's) of ∆ 1 − ∆ 2 and ∆ 3 , even when they are within the unitarity bounds. 6 This issue is easily solved by replacing D 1,2 → ( D 1,2 + D 1,2 ) in eq.(5.11), as suggested by the recursion relation (5.6), and by defining an improved differential basis A similar basis for parity odd structures is given by In practical computations it is more convenient to use the differential basis rather than the recursion relations and, if necessary, use the transformation matrix to rotate the results back to the ordinary basis. We have explicitly constructed the improved differential basis (5.12) and (5.13) up to l 1 = l 2 = 2. The rank of the transformation matrix depends on ∆ 3 and l 3 for any value of l 3 , and never vanishes, for any value of ∆ 3 allowed by the unitary bound. 7

Mixed Symmetry Exchanged Operators
In this subsection we consider correlators with two traceless symmetric and one mixed symmetry operator O (l 3 ,l 3 ) , with l 3 −l 3 = 2δ, with δ an integer. A correlator of this form has ∆l = 2δ and according to the analysis of section 2, any of its tensor structures can be expressed in a form containing an overall number δ of K i,jk 's if δ > 0, or K i,jk 's if δ < 0. We consider in the following δ > 0, the case δ < 0 being easily deduced from δ > 0 by means of a parity transformation. The analysis will proceed along the same lines of subsection 5.1. We first show a convenient parametrization for the tensor structures of the correlator, then we prove by deriving recursion relations how all tensor structures can be reached starting from a single seed, to be determined, and finally present a differential basis. We first consider the situation where l 3 ≥ l 1 + l 2 − δ and then the slightly more involved case with unconstrained l 3 .

Recursion Relations for l
It is convenient to look for a parametrization of the tensor structures which is as close as possible to the one (5.3) valid for δ = 0. When l 3 ≥ l 1 + l 2 − δ, any tensor structure of the correlator contains enough J 3,12 's invariants to remove all possible K 3,12 's invariants using the identity J 3,12 K 3,12 = 2I 31 K 1,23 − 2I 32 K 2,31 . (5.14) There are four possible combinations in which the remaining K 1,23 and K 2,31 invariants can enter in the correlator: K 1,23 I 23 , K 1,23 I 21 I 13 and K 2,31 I 13 , K 2,31 I 12 I 23 . These structures are not all independent. In addition to eq.(5.14), using the two identities 2I 12 K 2,31 = J 1,23 K 1,23 + 2I 13 K 3,12 , we can remove half of them and keep only, say, K 1,23 I 23 and K 2,31 I 13 . The most general tensor structure can be written as  The condition in m 12 derives from the fact that, using eqs.(5.15), one can set m 12 to zero in the tensor structures with p = 0, δ, see below. Attention should be paid to the subscript p. Structures with no subscript refer to purely traceless symmetric correlators, while those with the subscript p refer to three-point functions with two traceless symmetric and one mixed symmetry field. All tensor structures are classified in terms of δ + 1 classes, parametrized by the index p in eq.(5.16). The parity odd structures of traceless symmetric correlators do not enter, since they can be reduced in the form (5.16) by means of the identities (5.15). The class p exists only when l 1 ≥ p and l 2 ≥ δ − p. If l 1 + l 2 < δ, the entire correlator vanishes. Contrary to the symmetric traceless exchange, there is no obvious choice of seed that stands out. The allowed correlator with the lowest possible spins in each class, l 1 = p, l 2 = δ − p, m ij = 0, can all be seen as possible seeds with a unique tensor structure. Let us see how all the structures (5.16) can be iteratively constructed using the operators defined in section 4 in terms of the δ + 1 seeds. It is convenient to first construct a redundant basis where m 12 = 0 for any p and then impose the relation that leads to the independent basis (5.16). The procedure is similar to that followed for the traceless symmetric exchange. We first construct all the tensor structures with m 13 = m 32 = 0 for any spin l 1 and l 2 , and any class p, using the following relations: Notice that the operators D 1,2 and D 1,2 relate nearest neighbour classes and the iteration eventually involves all classes at the same time. The action of the D 2 and D 2 derivatives can be obtained by replacing 1 ↔ 2, p ↔ (δ − p) in the coefficients multiplying the structures and p + 1 → p − 1 in the subscripts, and by changing sign on one side of the equation. Structures with non-vanishing m 13 and m 23 are obtained using together with the corresponding relation with 1 ↔ 2 and p → p + 1. All the structures (5.16) are hence derivable from δ + 1 seeds by acting with the operators D 1,2 , D 1,2 and (I 12 I 21 ). The seeds, on the other hand, are all related by means of the following relation: We conclude that, starting from the single seed correlator with p = 0,  namely the three-point function of a scalar, a spin δ traceless symmetric operator and the mixed symmetry operator with spin (l 3 + 2δ, l 3 ), we can obtain all tensor structures of higher spin correlators.
Let us now see how the constraint on m 12 in eq.(5.17) arises. When p = 0, δ, namely when both K 1 and K 2 structures appear at the same time, combining eqs.(5.15), the following relation is shown to hold: (5.24) Using it iteratively, we can reduce all structures with p = 0, δ to those with m 12 = 0 and with p = 0, δ, any m 12 . 8 This proves the validity of eq.(5.16). As a further check, we have verified that the number of tensor structures obtained from eq.(5.16) agrees with those found from eq.(3.38) of ref. [21].

Recursion Relations for general l 3
The tensor structures of correlators with l 3 < l 1 + l 2 − δ cannot all be reduced in the form (5.16), because we are no longer ensured to have enough J 3,12 invariants to remove all the K 3,12 's by means of eq.(5.14). In this case the most general tensor structure reads (5.25) with p = 0, . . . , δ, q = 0, . . . , δ, p − q ≥ 0 and The parameter η in eq.(5.26) is necessary because the tensor structures involving K 3,12 (i.e. those with p = q) are independent only when j 3 = 0, namely when the traceless symmetric structure does not contain any J 3,12 invariant. All the tensor structures (5.25) can be reached starting from the single seed with p = 0, q = 0, l 1 = 0, l 2 = δ and m ij = 0. The analysis follows quite closely the one made for l 3 ≥ l 1 + l 2 − δ, although it is slightly more involved. As before, it is convenient to first construct a redundant basis where m 12 = 0 for any p, q and we neglect the factor η above, and impose only later the relations that leads to the independent basis (5.25). We start from the structures with p = q, which are the same as those in eq.(5.16): first construct the structures with m 13 = m 23 = 0 by applying iteratively the operators D 1,2 + D 1,2 , and then apply D 1 and D 2 to get the structures with non-vanishing m 13 and m 23 . Structures with p = q appear when acting with D 1 and D 2 . We have: The action of D 2 is obtained by exchanging 1 ↔ 2 and δ − p ↔ q in the coefficients multiplying the structures and replacing the subscript (p + 1, p) with (p, p − 1). For m 13 + m 23 < l 3 the first term in eq.(5.27) is redundant and can be expressed in terms of the known structures with p = q. An irreducible structure is produced only when we reach the maximum allowed value m 13 + m 23 = l 3 , in which case the third term in eq.(5.27) vanishes and we can use the equation to get the irreducible structures with p = q. Summarizing, all tensor structures can be obtained starting from a single seed upon the action of the operators D 1,2 , (D 1,2 + D 1,2 ), I 12 I 21 and R.

Differential Basis
A differential basis that is well defined for any value of l 1 , l 2 , l 3 and δ is   where ∆ ′ 1 = ∆ 1 +l 1 +m 23 −m 13 +n 2 −n 1 −p+q, ∆ ′ 2 = ∆ 2 +l 2 +m 13 −m 23 +n 1 −n 2 +2q −δ, and all parameters are defined as in eq.(5.26). The recursion relations found above have shown that the differential basis (5.28) is complete. One can also check that its dimensionality matches the one of the ordinary basis for any l 1 , l 2 , l 3 and δ. Like in the purely traceless symmetric case, the specific choice of operators made in eq.(5.28) seems to be enough to ensure that the determinant of the transformation matrix is non-vanishing regardless of the choice of ∆ 1 and ∆ 2 . We have explicitly checked this result up to l 1 = l 2 = 2, for any l 3 . The transformation matrix is always of maximal rank, except for the case l 3 = 0 and ∆ 3 = 2, which saturates the unitarity bound for δ = 1. Luckily enough, this case is quite trivial, being associated to the exchange of a free (2, 0) self-dual tensor [68] (see footnote 7). The specific ordering of the differential operators is a choice motivated by the form of the recursion relations, as before, and different orderings can be trivially related by using the commutators defined in eq.(4.8).

Computation of Four-Point Functions
We have shown in section 3 how relations between three-point functions lead to relations between CPW. The latter are parametrized by 4-point, rather than 3-point, function tensor structures, so in order to make further progress it is important to classify four-point functions. It should be clear that even when acting on scalar quantities, tensor structures belonging to the class of 4-point functions are generated. For example D 1 U = −U J 1,24 . We postpone to another work a general classification, yet we want to show in the following subsection a preliminary analysis, enough to study the four fermion correlator example in subsection 7.1.

Tensor Structures of Four-Point Functions
In 6D, the index-free uplift of the four-point function (3.1) reads g n (U, V ) T n (S 1 ,S 1 , .., S 4 ,S 4 ), (6.1) where T n are the 6D uplifts of the tensor structures appearing in eq.(3.1). The 6D kinematic factor K 4 and the conformally invariant cross ratios (U, V ) are obtained from their 4D counterparts by the replacement x 2 ij → X ij in eqs.(3.2) and (3.3). The tensor structures T n are formed from the three-point invariants (2.5)-(2.8) (where i, j, k now run from 1 to 4) and the following new ones: where i = j = k = l = 1, 2, 3, 4; K i,jkl and K i,jkl are totally anti-symmetric in the last three indices and the normalization factor is given by The invariants J ij,kl satisfy the relations J ij,kl = −J ij,lk + 2I ij . Given that, and the 4D parity transformations K i,jkl P ←→ K i,jkl and J ij,kl P ←→ −J ji,lk , a convenient choice of index ordering in J ij,kl is (i < j, k < l) and (i > j, k > l). Two other invariants H ≡ S 1 S 2 S 3 S 4 andH ≡ S 1S2S3S4 formed by using the epsilon SU (2, 2) symbols, are redundant. For instance, one has X 12 H = K 2,14 K 1,23 − K 1,24 K 2,13 .
Any four-point function can be expressed as a sum of products of the invariants (2.5)-(2.8) and (6.2)-(6.4). However, not every product is independent, due to several relations between them. Leaving to a future work the search of all possible relations, we report in Appendix A a small subset of them. Having a general classification of 4-point tensor structures is crucial to bootstrap a four-point function with non-zero external spins. When we equate correlators in different channels, we have to identify all the factors in front of the same tensor structure, thus it is important to have a common basis of independent tensor structures.

Counting 4-Point Function Structures
In absence of a general classification of 4-point functions, we cannot directly count the number N 4 of their tensor structures. However, as we already emphasized in ref. [21], the knowledge of 3-point functions and the OPE should be enough to infer N 4 by means of eq.(3.5). In this subsection we show how to use eq.(3.5) to determine N 4 , in particular when parity and permutation symmetries are imposed.
If the external operators are traceless symmetric, the CPW can be divided in parity even and odd. This is clear when the exchanged operator is also traceless symmetric: W The numbers N 4+ and N 4− in eq.(6.6) are always integers, because in the sum over r one has to consider separately r = (l,l) and r = (l, l), 9 and which give an equal contribution that compensates for the factor 1/2. When some of the external operators are equal, permutation symmetry should be imposed. We consider here only the permutations 1 ↔ 3, 2 ↔ 4 and 1 ↔ 2, 3 ↔ 4 that leave U and where again in the sum over r one has to consider separately r = (l,l) and r = (l, l).

Relation between "Seed" Conformal Partial Waves
Using the results of the last section, we can compute the CPW associated to the exchange of arbitrary operators with external traceless symmetric fields, in terms of a set of seed CPW, schematically denoted by W where D (p) 12 schematically denotes the action of the differential operators reported in the last section, and D (q) 34 are the same operators for the fields at X 3 and X 4 , obtained by replacing 1 → 3, 2 → 4 everywhere in eqs.(4.4)-(4.7) and (4.13). For simplicity we do not report the dependence of W on U, V , and on the scaling dimensions of the external and exchanged operators. The seed CPW are the simplest among the ones appearing in correlators of traceless symmetric tensors, but they are not the simplest in general. These will be the CPW arising from the four-point functions with the lowest number of tensor structures with a non-vanishing contribution of the field O l+2δ,l in some of the OPE channels. Such minimal four-point functions are 10 with just N seed 4 (δ) = 2δ + 1 (6.10) tensor structures. In the s-channel  operators O l+n,l , with −2δ ≤ n ≤ 2δ, are exchanged. We denote by W seed (δ) and W seed (δ) the single CPW associated to the exchange of the fields O l+2δ,l and O l,l+2δ in the four-point function (6.9). They are parametrized in terms of 2δ + 1 conformal blocks as follows (G In contrast, the number of tensor structures in grows rapidly with δ. Denoting it by N 4 (δ) we have, using eq.(6.6) of ref. [21]: It is important to stress that a significant simplification occurs in using seed CPW even when there is no need to reduce their number, i.e. p = q = 1. For instance, consider the correlator of four traceless symmetric spin 2 tensors. The CPW W O l+8,l (2, 2, 2, 2) is unique, yet it contains 1107 conformal blocks (one for each tensor structure allowed in this correlator), to be contrasted to the 85 present in W O l+8,l (0, 4, 0, 4) and the 9 in W seed (4)! We need to relate in order to be able to use the results of section 5 together with W seed (δ). As explained at the end of Section 4, there is no combination of first-order operators which can do this job and one is forced to use the operator (4.13): 13) where 11 c −1 n = 2(1 − n + 2δ) 2(n + 1) + δ + l + ∆ 1 − ∆ 2 + ∆ . (6.14) Equation (6.13) implies the following relation between the two CPW: where c 12 n = c n in eq.(6.14), c 34 n is obtained from c n by exchanging 1 → 3, 2 → 4 and the scaling dimensions of the corresponding external operators are related as indicated in eq.(6.13).
Summarizing, the whole highly non-trivial problem of computing W (p,q) O l+2δ,l (l 1 , l 2 , l 3 , l 4 ) has been reduced to the computation of the 2 × (2δ + 1) conformal blocks G O l+2δ,l (l 1 , l 2 , l 3 , l 4 ). 11 Notice that the scalings dimension ∆1 and ∆2 in eq.(6.14) do not exactly correspond in general to those of the external operators, but should be identified with ∆ ′ 1 and ∆ ′ 2 in eq.(5.28). It might happen that the coefficient cn vanishes for some values of ∆1 and ∆2. As we already pointed out, there is some redundancy that allows us to choose a different set of operators. Whenever this coefficient vanishes, we can choose a different operator, e.g. D1 → D1.

Examples
In this section we would like to elucidate various aspects of our construction. In the subsection 7.1 we give an example in which we deconstruct a correlation function of four fermions. We leave the domain of traceless symmetric external operators to show the generality of our formalism. It might also have some relevance in phenomenological applications beyond the Standard Model [39]. In the subsection 7.2 we consider the special cases of correlators with four conserved identical operators, like spin 1 currents and energy momentum tensors, whose relevance is obvious. There we will just outline the main steps focusing on the implications of current conservations and permutation symmetry in our deconstruction process.

Four Fermions Correlator
Our goal here is to deconstruct the CPW in the s-channel associated to the four fermion corre- For simplicity, we takeψ andχ to be conjugate fields of ψ and χ, respectively, so that we have only two different scaling dimensions, ∆ ψ and ∆ χ . Parity invariance is however not imposed in the underlying CFT. The correlator (7.1) admits six different tensor structures. An independent basis of tensor structures for the 6D uplift of eq. O l,l . The traceless symmetric CPW are obtained as usual by relating the three point function of two fermions and one O l,l to that of two scalars and one O l,l . This relation requires to use the operator (4.13). There are two tensor structures for l ≥ 1: where κ = 2 4∆ − (∆ + l) 2 , the superscript n in Φ indicates the shift in the scaling dimensions of the field and the operator O l,l is taken at X 0 . Plugging eq.(7.3) (and the analogous one for X andX ) in eq.(3.9) gives the relation between CPW. In order to simplify the equations, we report below the CPW in the differential basis, the relation with the ordinary basis being easily determined from eq.(7.3): where D 3 and D 4 are obtained from D 1 and D 2 in eq.(4.4) by replacing 1 → 3 and 2 → 4 respectively. The superscripts indicate again the shift in the scaling dimensions of the external operators. As in ref. [9] the CPW associated to the exchange of traceless symmetric fields is entirely determined in terms of the single known CPW of four scalars W seed (0). For illustrative purposes, we report here the explicit expressions of W (1,2) O l,l : where G (0) 0 are the known scalar conformal blocks [4,5]. It is worth noting that the relations (A.3)-(A.10) have to be used to remove redundant structures and write the above result (7.5) in the chosen basis (7.2).
The analysis for the mixed symmetry CPW W O l+2,l and W O l,l+2 is simpler. The three point function of two fermions and one O l,l+2 field has a unique tensor structure, like the one of a scalar and a (2, 0) field F . One has and similarly for the conjugate (0, 2) fieldF . Using the above relation, modulo an irrelevant constant factor, we get where W seed (1) and W seed (1) are defined in eq.(6.11). Explicitly, one gets

Conserved Operators
In this subsection we outline, omitting some details, the deconstruction of four identical currents and four energy-momentum tensor correlators, which are among the most interesting and universal correlators to consider. In general, current conservation relates the coefficients λ s of the three-point function and reduces the number of independent tensor structures. Since CPW are determined in terms of products of two 3-point functions, the number of CPW W O associated to external conserved operators is reduced with respect to the one of CPW for non-conserved operators W O : The coefficients Fp p 12O and Fq q 34Ō depend in general on the scaling dimension ∆ and spin l of the exchanged operator O. They can be determined by applying the operator defined in eq.(2.15) to 3-point functions.

Spin 1 Four-Point Functions
In any given channel, the exchanged operators are in the (l, l), (l + 2, l), (l, l + 2), (l + 4, l) and (l, l + 4) representations. The number of 3-point function tensor structures of these operators with the two external vectors and the total number of four-point function structures is reported in table 1. Each CPW can be expanded in terms of the 70 tensor structures for a total of 4900 scalar conformal blocks as defined in eq.(3.6). Using the differential basis, the 36 × 70 = 2520 conformal blocks associated to the traceless symmetric CPW are determined in terms of the single known scalar CPW [9]. The 16 × 70 = 1120 ones associated to O l+2,l and O l,l+2 are all related to the two CPW W seed (1) and W seed (1). Each of them is a function of 3 conformal blocks, see eq.(6.11), for a total of 6 unknown. Finally, the 2 × 70 = 140 conformal blocks associated to O l+4,l and O l,l+4 are expressed in terms of the 5 × 2 = 10 conformal blocks coming from the two CPW W seed (2) and W seed (2).
Let us see more closely the constrains coming from permutation symmetry and conservation. For l ≥ 2, the 5 + + 1 − tensor structures of the three-point function V 1 V 2 O l,l , for distinct nonconserved vectors, reads  Table 1: Number of independent tensor structures in the 3-point function V 1 V 2 O l,l when min(l,l) ≥ 2 − δ. In the last column we report N 4 as computed using eqs.(6.6) and (6.7) for general four spin 1, identical four spin 1 and identical conserved currents respectively. Subscripts + and -refers to parity even and parity odd structures. For conjugate fields we have where Fp p 12O are the coefficients entering eq.(7.10). The number of independent tensor structures is reduced from 6 to 2 + when l is even and from 6 to 1 − when l is odd, as indicated in the table 1. When ∆ = l + 4, eq.(7.12) is modified, but the number of constraints remains the same. The 3-point function structures obtained, after conservation and permutation is imposed, differ between even and odd l. Therefore, we need to separately consider the even and odd l contributions when computing N 4 using eq.(6.7). For four identical conserved currents, N 4 = 7 + , as indicated in table 1, and agrees with what found in ref. [49].

Spin 2 Four-Point Functions
The exchanged operators can be in the representations (l+2δ, l) and (l, l+2δ) where δ = 0, 1, ..., 4. The number of tensor structures in the three-point functions of these operators with two external spin 2 tensors is shown in table 2. We do not list here the number of CPW and conformal blocks for each representation, which could be easily derived from table 2. In the most general case of four distinct non conserved operators, no parity imposed, one should compute 1107 2 ∼ 10 6 conformal blocks, that are reduced to 49 using the differential basis, W seed (δ) and W seed (δ).
The constraints coming from permutation symmetry and conservation are found as in the spin 1 case, but are more involved and will not be reported. For four identical spin 2 tensors,  Table 2: Number of independent tensor structures in the 3-point function T 1 T 2 O l,l when min(l,l) ≥ 4−δ. In the last column we report N 4 as computed using eqs.(6.6) and (6.7) for general four spin 2, identical four spin 2 and energy momentum tensors respectively. Subscripts + and − refers to parity even and parity odd structures. For conjugate fields we have N 12 O(l,l+δ) = N 12 O(l+δ,l) .
namely for four energy momentum tensors, using eq.(6.7) one immediately gets N 4 = 22 + + 3 − , as indicated in table 1. The number of parity even structures agrees with what found in ref. [49], while to the best of our knowledge the 3 parity odd structures found is a new result. Notice that even if the number of tensor structures is significantly reduced when conservation is imposed, they are still given by a linear combination of all the tensor structures, as indicated in eq.(7.10). It might be interesting to see if there exists a formalism that automatically gives a basis of independent tensor structures for conserved operators bypassing eq.(7.10) and the use of the much larger basis of allowed structures.

Conclusions
We have introduced in this paper a set of differential operators, eqs.(4.4), (4.5) (4.6) and (4.13), that enables us to relate different three-point functions in 4D CFTs. The 6D embedding formalism in twistor space with an index free notation, as introduced in ref. [11], and the recent classification of three-point functions in 4D CFTs [21] have been crucial to perform this task. In particular, three-point tensor correlators with different tensor structures can always be related to a three-point function with a single tensor structure. Particular attention has been devoted to the three point functions of two traceless symmetric and one mixed tensor operator, where explicit independent bases have been provided, eqs.(5.25) and (5.28). These results allow us to deconstruct four point tensor correlators, since we can express the CPW in terms of a few CPW seeds. We argue that the simplest CPW seeds are those associated to the four point functions of two scalars, one O 2δ,0 and one O 0,2δ field, that have only 2δ + 1 independent tensor structures.
We are now one step closer to bootstrapping tensor correlators in 4D CFTs. There is of course one important task to be accomplished: the computation of the seed CPW. One possibility is to use the shadow formalism as developed in ref. [11], or to apply the Casimir operator to the above four point function seeds, hoping that the second order set of partial differential equations for the conformal blocks is tractable. In order to bootstrap general tensor correlators, it is also necessary to have a full classification of 4-point functions in terms of SU (2, 2) invariants. This is a non-trivial task, due to the large number of relations between the four-point function SU (2, 2) invariants. A small subset of them has been reported in the appendix A but many more should be considered for a full classification. We hope to address these problems in future works.
We believe that universal 4D tensor correlators, such as four energy momentum tensors, might no longer be a dream and are appearing on the horizon!