R-current three-point functions in 4d $\mathcal{N}=1$ superconformal theories

In 4d $\mathcal{N}=1$ superconformal field theories (SCFTs) the R-symmetry current, the stress-energy tensor, and the supersymmetry currents are grouped into a single object, the Ferrara-Zumino multiplet. In this work we study the most general form of three-point functions involving two Ferrara-Zumino multiplets and a third generic multiplet. We solve the constraints imposed by conservation in superspace and show that non-trivial solutions can only be found if the third multiplet is R-neutral and transforms in suitable Lorentz representations. In the process we give a prescription for counting independent tensor structures in superconformal three-point functions. Finally, we set the Grassmann coordinates of the Ferrara-Zumino multiplets to zero and extract all three-point functions involving two R-currents and a third conformal primary. Our results pave the way for bootstrapping the correlation function of four R-currents in 4d $\mathcal{N}=1$ SCFTs.

Appendix D. Conventions for the supersymmetric derivatives 39

Introduction
In the last decade the conformal bootstrap has been widely used to explore the space of conformal field theories (CFTs), both from a numerical perspective [1] but also from an analytical one [2]. Important results have been obtained for 3d condensed matter systems, but also for higher-dimensional theories in presence of supersymmetry (see [3] for a review and a summary of important results).
Most of these techniques heavily rely on the computation of conformal blocks, or superconformal blocks in the case of superconformal field theories (SCFTs). While for CFTs this problem has now been completely solved in 3d [4] and 4d [5,6], it is still an open question in several supersymmetric cases. Superconformal blocks can be expressed as finite linear combinations of ordinary blocks. Nevertheless, finding the exact expression is technically challenging. A notable example are the superconformal blocks of the stress-energy tensor multiplet four point function in 4d N = 2 theories, see e.g. [7].
In this work we focus on 4d N = 1 theories. Computations of superconformal blocks have already been performed in the literature for four-point functions of scalar operators, e.g. chiral or antichiral [8], linear [9,10], and general [11,12]. Here we take the next logical step and address a more complicated case, i.e. we compute all necessary ingredients for the calculation of superconformal blocks of four-point functions involving the R-current J µ . The vector operator J µ is in the same multiplet as the supersymmetry current and the stress-energy tensor, called the Ferrara-Zumino multiplet, J µ [13]. Our results are obtained by explicitly working out the projection of the superconformal three-point function with the Ferrara-Zumino multiplet at the first two points and a general allowed operator at the third. The main motivation is to use these results to bootstrap the four-point function of J µ . This will provide a new way to explore the space of SCFTs, and hopefully shed more light on the "minimal" 4d N = 1 SCFT studied with bootstrap techniques in [12,14] and attempted to be identified by analytical means in [15].
Unlike the case of extended supersymmetry in 4d, in N = 1 SCFTs the supermultiplet containing the stress-energy tensor does not contain a scalar primary operator. Dealing with spinning operators raises the complication of the computations significantly, although the general procedure remains the same.
Let us outline the logic we follow here. Our starting point is the superconformal three-point function in superspace with the Ferrara-Zumino multiplet at the first two points and a general allowed superconformal multiplet at the third. 1 The superspace expression for the three-point function, constructed following the constraints laid out in [16,17], involves many structures with a priori independent coefficients. The first step is to work out the relations among these coefficients due to the shortening condition satisfied by the Ferrara-Zumino multiplet. Typically, this reduces the number of independent coefficients drastically, and in some cases sets the whole three-point function to zero. Subsequently, we perform an expansion in the fermionic coordinates θ 3 andθ 3 , after setting θ 1 = θ 2 = 0 andθ 1 =θ 2 = 0 in order to focus on the R-current at the first two points. With this expansion we are able to identify three-point functions of conformal primary operators.
This last step is the most complicated and delicate one: any given order of the expansion contains a combination of conformal primaries and descendants which must be disentangled. For example, at order θ 3θ3 the expansion of the superconformal three-point function contains not only terms belonging to three-point functions of the schematic form JJ(QQO) p , where Q is the supersymmetric charge and "p" denotes that the operator is primary, but also terms belonging to three-point functions of the schematic form JJ(P O) , where P is the generator of translations.
The latter contributions can be subtracted away using the results of [18], where the specific way contamination from conformal descendants can happen was worked out in generality. To carry out our calculations we have expanded the Mathematica package 2 developed for the purposes of [18].
For the structures associated with three-point functions of conformal primary operators we have used the Mathematica package CFTs4D [19].
A non-trivial check on our computations is supplied by the fact that when the third operator in the three-point function satisfies a shortening condition, then a unitarity bound is saturated and the corresponding three-point function should vanish. This typically happens automatically after the Ward identities for conservation at the first two points have been solved, i.e. the solution for the independent three-point function coefficients involves explicit factors of ∆ − ∆ u , where ∆ u is the dimension at the unitarity bound. While in some of our cases this story is repeated, we have also encountered situations where solving the Ward identities at the first two points is not enough to guarantee vanishing of the three-point function when the third operator saturates its unitarity bound. The Ward identity at the third point needs to be imposed in those cases, something that results in the proper vanishing of the three-point function. In some cases the resulting requirement is non-trivial, i.e. it does not set all independent (after satisfying the Ward identity at the first two points) coefficients to zero at the unitarity bound, but rather it relates them in the appropriate way.
In Sec. 2 we review known results about the structure of three-point functions with two conserved spin-one currents at the first two points. In Sec. 3 we explore general constraints on our three-point functions of interest in superspace. We introduce an index-free notation and we provide counting arguments for the number of independent structures in superconformal three-point functions with two Ferrara-Zumino and a general multiplet in N = 1 superspace.  [20] we lift the fields to 6d and we introduce 6d spinors S iα andSα i , i = 1, 2, 3, to contract the spacetime indices. The three-point function is non-zero in three cases: where the prefactor is and we have defined the tensor structures Permutation symmetry and conservation of the current J µ impose a set of conditions summarized in Table 1. We found two special cases: when k = 0, ℓ = 0, then λ (4) = 0 since the associated tensor structure does not exist. As a consequence, λ (2) vanishes as well and there is only one degree of freedom. A second exception is for k = 2, ℓ = 0; in this case permutation symmetry and current conservation sets the three-point function to zero, expect for the special case ∆ = 2, when λ (3) = λ (4) , while all the rest vanishes. However, in SCFTs this operator is below the unitarity bounds (see Sec. 4). Besides these special cases, we stress that all the denominators in Table 1 are non-zero whenever the dimension of O ℓ+k,ℓ satisfies the unitarity bounds [21] ∆ ≥ ℓ + 1 2 k + 2 for ℓ > 0 ,

Three-point function of two Ferrara-Zumino and a general multiplet
Let us now move on to the supersymmetric case. For our analysis we will mostly follow the formalism developed in [16,17] and we follow the conventions of Wess and Bagger [22]. With N = 1 supersymmetry the conserved currents arising from superconformal transformations are 3 Schematically we have: Xij = Xi · Xj , Iij = SiSj, J i,jk = SiXj X k Si/X jk , K i,jk = Sj XiS k X jk /(Xij X ik ). In addition, X ab is obtained contracting X M with the 6d gamma matrices. Finally J1 ≡ J1,23, J2 ≡ J2,13, J3 ≡ J3,12 and similarly for Ki. In CFTs4D the structures I, J, K are denotedÎ,Ĵ,K.

General properties
In this section we study the most general form of the three-point function of two Ferrara-Zumino multiplets and a third general with 4d N = 1 superconformal symmetry. We recall that superconformal multiplets are labelled by two integers, j and, indicating that the superconformal primary in the multiplet transforms in the ( 1 2 j, 1 2 ) representation of the Lorentz group, and two reals, q andq, which give the scaling dimension and R-charge of the superconformal primary operator via While the supercurrent satisfies q J =q J = 3 2 , for a general supermultiplet O the values q,q can assume any value consistent with the unitarity bounds 5 [24]: Our goal is to start from the superspace expression of the three-point function Before analyzing each of the cases (A)-(C) individually, let us discuss the general properties of the correlator (3.6). As shown in [17], the most general three-point function consistent with superconformal symmetry can be written as withθ ij =θ i −θ j and the supersymmetric interval between x i and x j defined by 6 In this work we use interchangeably the terminology "shortening condition" and "Ward indentity". 7 Examples of this have appeared before in the literature [10,12]. 8 In [17] X, Θ and Θ correspond to, respectively, X3, Θ3 and Θ3.
This counting is performed in the following section. After that we construct a complete basis for the cases of interest-namely three-point functions of operators with vanishing R-charge-by providing an equal number of independent tensor structures. Their independence can be easily proven by setting to zero the Grassmann coordinates θ 1,2 andθ 1,2 and matching with the nonsupersymmetric three-point functions reviewed in Sec. 2. The tensor structures associated to three-point functions of operators with non-zero R-charge can be read from the Mathematica notebook attached to this submission. They are constructed in the same way, and their number agrees with the counting of the next section as well. We also checked their linear independence by replacing numerical values for the various quantities that appear.

Counting supersymmetric tensor structures
In this section we obtain a group theoretical counting of the independent number of tensor structures appearing in (3.15) along the same lines as [25].
Let us analyze first the case of an operator O j, with zero R-charge. We can start by dividing the function t(η i ,η i , X, Θ, Θ) into three parts. The first part contains neither Θ nor Θ; it is thus built with [ij], [i], [ī] only. The second part is analogous to the previous one but with an overall Θ 2 Θ 2 factor. The third part is instead built with exactly one Θ and one Θ. In order to enumerate the structures in the first part we can simply follow a standard approach for non-supersymmetric CFTs. One possible way is to choose a conformal frame [25,26] that fixes all bosonic coordinates and breaks Spin(2, 4) → Sp(2, R). After restricting the polarizations η i andη i to this subgroup, η iα andη iα ≡ X ααηα i transform in the same representation. Therefore we are allowed to make the contractions 10 invariant tensors built out of η i andη i , which are in one-to-one correspondence with the tensor The second part presents no difference apart from the trivial Θ 2 Θ 2 overall factor. The third part, instead, can be interpreted in the following way: since there is only one Θ and only one Θ we can ignore the fact that they anticommute and replace them by a fourth pair of polarizations parentheses denote symmetrization. In the index-free notation this condition is enforced by contracting all indices with the same η. Tensors not corresponding to irreducible representations must be contracted with independent polarizations. For example, the following operator belongs Now we are ready to perform the actual counting. In order to do so we will use the main formula derived in [25] N = Res .
(3.22) 10 The contractions are made with ǫ αβ and ǫ αβ . All indices are undotted at this point. 11 The choice of attaching the polarizations η4,η4 to the third operator is arbitrary and does not affect the result. In this case it is convenient because we want to keep manifest the permutation symmetry in the first two points.
Here ρ 1 = ρ 2 = ( 1 2 , 1 2 ) and ρ 3 is the spin representation of the third operator. The notation Res G H indicates the restriction of a representation of G to a representation of H ⊆ G, the superscript (ρ) H denotes the H-singlets in ρ. We assume that this formula generalizes for ρ 3 not irreducible.
Moreover, as remarked in [25], Res G H commutes with the tensor product. A last ingredient is necessary, namely the permutation symmetry of the first two points. This is taken care of in [25] as well. It is sufficient to replace ρ 1 ⊗ ρ 2 by S 2 ρ 1 if j is even and by ∧ 2 ρ 1 if j is odd, where S 2 and ∧ 2 denote respectively the symmetrized square and the exterior square of representations.
Assuming, now, j − even, 12 we can write down the formulae where we abbreviated Res SO (1,2) with Res and a superscript SO(1, 2) in all terms is understood. We denote the number of independent structures in the three-point function J 1,1 J 1,1 O j, with N (j,). The factor of "2" counts the first and second part. The second term comes from putting ) and corresponds to the third part. Notice that when the first term has the S 2 product the second has the ∧ 2 product and vice versa. This is because the tensor product ). The result can be computed with the well known relations where ℓ indicates the spin-ℓ representation of SO (1,2). Finally the (anti)symmetrized products are given by where S 2 and ∧ 2 inside the parenthesis ( 1 2 j, 1 2 ), stand for the direct sum of all possible pairs of the resulting irreps.
Collecting all the above results, the numbers of independent tensors structures consistent with 12 If j − is odd the result is trivially zero. permutation symmetry read Let us now consider the case of a supermultiplet O j, with non-zero R-charge. In this case the superconformal primary does not contribute to the three-point function. As a consequence the structures in t(η i ,η i , X, Θ, Θ) contain an overall Θ, Θ, Θ 2 or Θ 2 . If the R-charge is ±2 the problem is readily solved by multiplying the non-supersymmetric three-point functions by, respectively, Θ 2 or Θ 2 . The counting is therefore the same as in Sec. 2.
When instead the R-charge is ±1 (say 1 for simplicity) we can derive a similar formula as in Eq. (3.23). Here the structures can be divided into two parts, the first proportional to Θ and the second proportional to Θ 2 Θ. In both cases the free fermionic variable can be interpreted as an extra η 4 orη 4 contracting a reducible operator O belonging to, respectively, ( Again notice that the products S 2 and ∧ 2 are inverted in the two terms. The counting now gives representations ρ 1 , ρ 2 and ρ 3 respectively. The cases with non-trivial permutation symmetries can be treated similarly as above. Every function t(η i ,η i , X, Θ, Θ) can contain a subset of the Which ones are present depends on the R-charges of the operators O 1 , O 2 , O 3 , which we will call The possible values are δ = ±2, ±1, 0. Let us denote as N X (ρ 1 , ρ 2 , ρ 3 ) the number of structures of a given order X in Θ, Θ, where X is any monomial in (3.29). Following the analysis above we have where again Res ≡ Res SO(1,2) and a superscript SO (1,2) in all terms is understood. Then the general formula for the number N (ρ 1 , ρ 2 , ρ 3 ; δ) of tensor structures in the three-point function (3.32)

Conserved tensor structures
To conclude this section we will address the issue of conservation. In particular we will study the consequence of (3.1) on a general three-point function J J O . For simplicity we will omit the superspace coordinate dependence. The constraints we need to impose are These conditions are not independent; in fact there are linear relations between them. First we can observe that taking the derivative D at the second point of (3.33a) and the derivative D at the second point of (3.33b) give the same result, modulo permuting the first two operators, Moreover, by taking D of (3.33a) and permuting points z 1 and z 2 we obtain identically zero. The same holds if we take D of (3.33b). The prescription to count the number of conserved tensor structures [25] is to take the number of non-conserved tensor structures, subtract all degrees of freedom contained in the equations (3.33) and add back all linear relations between such equations.
The complication with supersymmetry is that a superspace equation decomposes into a certain number of ordinary bosonic equations by projecting on the various terms in (3.29). This depends on the R-charge of O. Let us start assuming that O is real. The conservation conditions impose a number of constraints equal to the number of tensor structures present in (3.33). This number is given by 14 Even though N Θ = N Θ 2 Θ , etc., we keep them distinct to track down the various contributions.
As anticipated, however, not all the tensor structures in (3.33) give a non trivial constraint. This is a consequence of the fact that the three-point functions D(3.33a) and D(3.33b) are made of identical operators. To keep this into account one must subtract from (3.36) the numbers Similarly, given the relation D(3.33a) ∼ D(3.33b), we should naîvely subtract from (3.36) the number However, the above expression would give rise to an over-counting: the conditions given by  In addition, since the currents J are identical, we need to take into account the permutation symmetry as we explained in the previous section by replacing the product ρ 1 ⊗ ρ 2 by either There is a subtlety in the (anti)symmetrization of two DJ 's or two DJ 's: these operators get an extra minus due to their fermionic nature. Thus for ℓ even (odd) we must take 14 We denote the various representations in NX (. . .) in the following way: where, as before, N (j, ) is a shorthand for N J , J , ( 1 2 j, 1 2 ); 0 . We can similarly obtain the respective formulas when O has non-zero R-charge. Without loss of generality we take the R-charge to be negative. 16 Skipping the details of the derivation we show the answer for R = −1, In all cases with non-zero R-charge (3.41) and (3.42) yield non-positive results. Therefore we conclude that there are no structures allowed after conservation, as anticipated in Sec. 3.1.

Ward identities and their solution
In this section we explore the consequences of the Ferrara-Zumino shortening conditions, Eq. (3.1).
For each of the cases (A)-(C) we write the correlator in superspace in terms of tensor structures satisfying the conditions (3.10), (3.13) and eventually (3.14). One can straightforwardly check their independence, and since they match in number the predictions of Sec. 3.3, they form a basis of superconformal three-point functions.

Case
We begin by considering the case where O is a spin-ℓ Lorentz representation with zero R-charge.

Even ℓ
When all Grassmann variables are set to zero, then there are four possible parity-even structures in t A for ℓ even [27]. The Ward identity that follows from (3.1) will relate these structures, leaving, in the end, at most two independent parity-even structures [27]. In our case there may also be structures that become identically zero when all Grassmann variables are set to zero.
It is easier to perform the Ward-identity analysis by first introducing auxiliary commuting spinors η i ,η i , i = 1, 2, 3, as in (3.16). We can use these spinors to write the three-point function in the form 17 with the definition As anticipated in Sec. 3.3, we can parametrize t A in terms of 16 coefficients, where the tensor structures P A survive, while all the others vanish. Hence, the coefficients A i must be related to the coefficients λ (j) introduced in Sec. 2 for the case of traceless symmetric tensors (k = 0). It is straightforward to find The Ward identities following from (3.1) are satisfied if Equation (4.6) decomposes into fourteen independent equations for the sixteen a priori indepen- 17 In order to have a consistent treatment of all cases we have chosen to express t (ℓ even) A as a function of U, ΘΘ instead of X, X. dent coefficients A 1 , . . . , C 8 . 18 We may express all coefficients in terms of A 1 and A 2 to find The above expression are divergent if we set ∆ = 2, however in unitary theories this is not an issue as long as ℓ > 0. The special value ℓ = 0 is discussed separately in Appendix C. In that case there is then only one undetermined coefficient, consistently with the results of [17].

Odd ℓ
For odd spins we may write where A i , B i , C i , D i , E i are real constants and the tensor structuresP In the lowest component of the three-point function (4.2) for general odd spin ℓ there is a parity-odd and a parity-even structure, with respective coefficients denoted by λ (−) and λ (2) in Sec. 2. In terms of A 1 and A 2 of (4.8) we find Again, the shortening condition (3.1) implies For generic ∆, ℓ this gives fourteen independent equations for the sixteen unknowns A 1 , . . . , E 6 , and thus there are two undetermined coefficients, just as in the even-spin case. If we choose A 1 and C 1 as independent we have The result A 2 = 0 is expected because for conserved currents there is only one parity-odd structure in t (ℓ odd) A if all Grassmann variables are set to zero [27]. If we set ℓ = 1 and ∆ = 3, corresponding to the third operator in the three-point function being the supercurrent, we find a singularity in the above expression. This is just an artifact of the choice of variables and is resolved in Appendix C. In the same Appendix we also show the relation between the coefficients defined here and the anomaly coefficients a and c.

Case B:
( 1 2 (ℓ + 2), 1 2 ℓ) operators Here we start with where O has zero R-charge. The unitarity bound is  The symmetry properties for the first two points is identical to (3.19). However, unlike in case (A), we do not have a reality condition as in (3.20).

Even ℓ
For general even ℓ, we can parametrize the correlator as where the structures P The relations between A 1 and A 2 of (4.15) and the λ (i) in (2.1) is The Ward identity this time requires two independent conditions, due to the lack of a reality condition on t B . This leads, for generic ∆ and ℓ, to nine linear constraints for the ten unknowns A 1 , . . . , C 6 . Solving in terms of A 1 we find (4.18)

Odd ℓ
For general odd ℓ we find the independent structures where the structuresP We now need to impose the conservation at the first two points. For generic ∆ and ℓ this gives eleven independent linear constraints for the thirteen unknowns A 1 , . . . , C 9 . Therefore there are two undetermined coefficients, which we choose to be A 1 and C 1 . The result is saturates the unitarity bound ((4.1) or (4.13) respectively). In the present case this is not true: gives rise to a further non-trivial constraint, which is solved if C 1 takes the form with C 1 ′ another arbitrary constant.

Even ℓ
In this case we parametrize the correlator in terms of four tensor structures t (ℓ even) C where as usual the quantities P (4.28) Similarly to (4.17) we need to require This leads, for generic ∆ and ℓ, to three linear constraints for the four unknowns A, . . . , C 2 .
Solving in terms of A we find Analogously to Sec. 4.2.2, conservation at points z 1,2 alone is not sufficient to ensure conservation at point z 3 when ∆ saturates the bound (4.25). Instead, the constraint implies that A must be of the form where A ′ is an arbitrary constant.

Odd ℓ
Similarly to the even-ℓ case we have the four structures where the structuresP (i) C are defined in Appendix A. This three-point function vanishes in the limit θ 3 ,θ 3 → 0, consistently with the fact that conformal invariance does not allow any structure for a three-point function of the form J(x 1 )J(x 2 )O ℓ+4, ℓ (x 3 ) when ℓ is odd.
The Ward identities for conservation at points z 1,2 impose three constraints on these four constants. Thus, choosing C 1 as independent, we obtain As in the even-ℓ case conservation at point z 3 is not automatic. We need to impose the constraint in (4.31) which amounts to requiring C 1 to be of the form

Three-point function coefficients
In this section we will set θ 1,2 =θ 1,2 = 0 and perform an expansion of the three-point functions presented in Sec. 4 in θ 3 ,θ 3 . The results will allow us to extract the OPE coefficients of the various operators inside the superconformal multiplet O j, (z) in terms of the coefficients the second order. The precise way in which this mixing takes place is described in [18]. Following those results we are able to isolate the contributions of each primary and compute their OPE coefficients in the basis for non-supersymmetric three-point functions adopted in [19] and reviewed in Sec. 2. The expansion of (4.2) in θ 3 can be performed using an extension of the Mathematica package developed in [18]. Notice that the operators (QQO) j±1,±1;p (x) and (Q 2 Q 2 O) j,;p (x) are not normalized in a standard way, but rather according to [18, Table 1].
In the following we will go through each of the cases presented in Sec. 4: in each subsection we explicitly write the non-vanishing three-point functions between the currents J and the various primary superconformal descendants in terms of the tensor structures introduced in Sec. 2. In order not to overload the notation we use the same symbols for the OPE coefficients multiple times: since each subsection corresponds to a different super-primary, we are confident that this will not create any confusion. Schematically, we indicate with λ (i) the OPE coefficients associated to the superconformal primary, λ  Table 1. Thus, we do not show the value of all of them, but only the independent ones and a few non-trivial relations. We have checked explicitly that all the constraints of Table 1  The latter case instead corresponds to a primary with j − = ±2: we expect therefore four tensor structures, parametrized by only one independent coefficient. More in detail we have where λ Using (4.7), we obtain the following relations: When the unitarity bound (4.1) is saturated we get λ As expected in (5.1) we do not get any contribution associated with a parity-even tensor structure. Also, the relation between λ ±∓ is exactly the one expected from Table 1 After imposing the Ward identity constraints (4.7) we obtain consistently with the relations of Table 1. All coefficients vanish at the unitarity bound (4.1), as a consequence of the multiplet shortening.

Odd ℓ
Just as in the even-spin case, at order θ 3θ3 we have four different three-point functions. Two of these three-point functions involve five structures (because they contain even-spin operators at the third point) parametrized by two independent coefficients, while the other two involve four structures parametrized by one coefficient [19]. The first three-point function takes the form and we also have The coefficients of these three-point functions are given by, after use of (4.11), −− = 0, as necessary due to multiplet shortening.
At order θ 2 3θ 2 3 and after subtraction of three-point functions involving descendants of O we are left with a single three-point function. This involves one independent coefficientλ (−) , multiplying the structure S (−) , exactly as in in (4.9) but with ∆ → ∆ + 2. We find 9) consistently both with the unitarity bound and the fact that there is no parity-even structure in the three-point function involving two conserved currents in the first two points when ℓ is odd.

Even ℓ
At order θ 3θ3 we have four different three-point functions; one of them is identically zero by conservation (see Table 1), another one involves a traceless symmetric tensor with odd spin, while the other two have one independent coefficient each. More specifically, we have as well as and When the unitarity bound (4.13) is saturated, λ −− = 0, as required by multiplet shortening.
At order θ 2 3θ 2 3 we expect four tensor structures parametrized by one independent coefficient. The structures are the same as the ones in (4.16) but with ∆ → ∆ + 2. If we denote then, after using (4.18), we obtain As expected allλ (j) go to zero when the bound (4.13) is saturated.

Odd ℓ
At order θ 3θ3 we have four different three-point functions. One of them is parametrized by only one coefficient, another is the three-point function of a symmetric traceless operator with even spin and contains therefore two independent coefficients, while the other two contain one coefficient each. We have and while the symmetric traceless one is With the use of (4.21) and (4.23) we obtain λ (1) where we have defined −− = 0, as required by multiplet shortening.
At order θ 2 3θ 2 3 we expect two structures parametrized by one coefficient. These structures are the same as the ones in (4.20) but with ∆ → ∆ + 2. More specifically, we have After using (4.21) and (4.23) we obtain where we defined the denominators (5.23) Consistently with multiplet shorteningλ (j) = 0 at the unitarity bound (4.13).

Even ℓ
At order θ 3θ3 only one three-point function is non-zero, namely With the use of (4.30) and (4.32) we find As expected from multiplet shortening, both coefficients vanish when ∆ saturates (4.25).
At order θ 2 3θ 2 3 we have only one structure, namely the same one multiplied by λ in (4.28) with ∆ → ∆ + 2, The coefficient is determined to bê which vanishes for ∆ saturating (4.25) as needed.

Odd ℓ
We have three non-zero three-point functions with one independent coefficient each, i.e.
With the use of (4.34) and (4.35) we obtain We checked that this is indeed the case.

Discussion
In this work we have studied the constraints imposed by N = 1 superconformal symmmetry on the three-point function involving two Ferrara-Zumino supermultiplets J αα and a third supermultiplet where we have omitted a kinematic prefactor, and the partial waves W O represent the contribution of an entire superconformal multiplet to the four-point function, In the above expression g (a,b;i) ∆ O ′ ,ℓ O ′ are the non-supersymmetric conformal blocks for spinning correlators computed in [5,6] and the λ's are the three-point function coefficients between two currents and the various components of the supermultiplet O; because of supersymmetry they are all determined in terms of the superconformal primary ones. The main result of this work is the exact form of these relations. This was achieved in Sec. 5. Notice that the coefficients presented there do not correspond exactly to the ones in (6.2) since the superconformal descendants do not have a standard normalization. Thus, one has to divide by their norms, which have been already computed [18].
As anticipated, the next step is to bootstrap numerically the correlation function of four Rcurrents, along the same line of the 3d global symmetry current bootstrap [28] and the 3d stressenergy tensor bootstrap [29]. In particular one has to remove the redundant crossing symmetry conditions by properly taking into account the conservation constraint. This step could be more involved in the supersymmetric case. Given the universal nature of the Ferrara-Zumino multiplet, this study will open a window on all local 4d N = 1 SCFTs, potentially leading to discovering new ones and hopefully shedding light on the putative minimal SCFT studied in [12,14].
The three-point functions computed in this work can also be used in combination with lightcone bootstrap techniques [2] and the recent OPE inversion formula [30] to study the behaviour of Regge trajectories in supersymmetry.
Finally, starting from our results in superspace, with a bit more technical effort it is possible to compute the three-point functions of other components of J , such as the stress-energy tensor.
Although at the level of three-point functions one does not get any additional information, it is not excluded that the crossing constraints coming from their four-point functions would impose independent restrictions. It would be interesting to investigate this direction.
In the case of odd spins and before application of the Ward identity for the supercurrent there is a parity-odd and a parity-even structure if all Grassmann variables are set to zero [27].

Appendix D. Conventions for the supersymmetric derivatives
In order to impose (3.1) we follow the formalism of [17] and pass the derivatives D 1α and D 1α through the prefactor in (3.7). This can be done using the identities D 1α (x 31 ) αα (x1 3 2 ) 2 F α (X, Θ, Θ) = −i The superspace derivatives are defined in the following way