Three-point functions of higher-spin spinor current multiplets in ${\mathcal N}=1$ superconformal theory

In this paper, we study the general form of three-point functions of conserved current multiplets $S_{\alpha(k)}= S_{(\alpha_1 \dots \alpha_k)}$ of arbitrary rank in four-dimensional ${\mathcal N}=1$ superconformal theory. We find that the correlation function of three such operators $\langle \bar{S}_{\dot{\alpha}(k)} (z_1) S_{\beta(k+l)} (z_2) \bar{S}_{\dot{\gamma}(l)} (z_3) \rangle$ is fixed by the superconformal symmetry up to a single complex coefficient though the precise form of the correlator depends on the values of $k$ and $l$. In addition, we present the general structure of mixed correlators of the form $\langle \bar{S}_{\dot{\alpha}(k)} (z_1) S_{\alpha(k)} (z_2) L(z_3) \rangle$ and $\langle \bar{S}_{\dot{\alpha}(k)} (z_1) S_{\alpha(k)} (z_2) J_{\gamma \dot{\gamma}} (z_3) \rangle$, where $L$ is the flavour current multiplet and $J_{\gamma \dot{\gamma}}$ is the supercurrent.

The most important conserved currents in conformal field theory are the energymomentum tensor and vector currents. In the supersymmetric case, they are embedded in special multiplet of conserved currents. The energy-momentum tensor is, therefore, replaced with the supercurrent [25] (see also [26][27][28][29][30]) and vector currents are replaced with flavour current multiplets [31]. Three-point functions of these current multiplets have been extensively studied. In general, (super)conformal field theories also possess higher-spin conserved currents. In the case of three-dimensional conformal field theory, it was proven (under certain assumptions) by Maldacena and Zhiboedov in [32] that all correlation functions of higher-spin currents are equal to the ones in a free theory. The theorem of Maldacena and Zhiboedov was later generalised by Stanev [33] and by Alba and Diab [34,35] to the four-(and higher-) dimensional case. One can view these results as the analog of the Coleman-Mandula theorem [36] for conformal field theories.
We believe that the analysis of [32][33][34][35] has some limitations. First, the authors of [32][33][34][35] considered only bosonic symmetric traceless currents. However, in supersymmetric theories conserved currents form supermultiplets consisting of both bosonic and fermionic component currents. Second, the results of [32][33][34][35] are proven under certain assumptions, the main being the existence of only one conserved current of spin two which is the energy-momentum tensor. In [32] it was shown that in three-dimensional conformal field theory the existence of a half-integer higher-spin conserved current implies the existence of one more conserved current of spin two. This means that in supersymmetric conformal field theory possessing higher-spin currents the assumptions of [32] might be violated. It is likely that the same conclusion also holds in four dimensions. Therefore, in the supersymmetric case it is unclear if higher-spin current multiplets exist only in free theory. In any case, regardless whether a theory is free or not, it is interesting to understand the general structure of their correlation functions.
In the non-supersymmetric case, the general structure of the three-point functions of conserved bosonic, vector currents of arbitrary spin was determined by Stanev [37] and Zhiboedov [38], see also [39] for similar results in the embedding formalism. The aim of this paper is to make first steps towards determining the three-point functions of conserved higher-spin currents in the N = 1 supersymmetric case in four dimensions. In fact, we will be interested in multiplets which do not contain vector currents. To start with, it is worth reminding the reader of the known conformal current multiplets. (1. 2) The case k = 1 was first considered in [19], where it was shown that the spinor supercurrent S α naturally originates from the reduction of the conformal N = 2 supercurrent [43] to N = 1 superspace.
• Finally, the k = l = 0 case corresponds to the flavour current multiplet [31], L =L, constrained by Do all of these multiplets occur in superconformal field theories?
The conformal supercurrent J αα exists in every N = 1 superconformal field theory. Flavour current multiplets exists in every superconformal field theory possessing an internal symmetry Lie group. The spinor current multiplet S α exists in every N = 2 superconformal field theory realised in terms of N = 1 superfields. Explicit realisations in terms of free conformal scalar multiplets are known for the conformal current higherspin multiplets J α(k)α(k) and J α(k+1)α(k) , with k > 1, both in Minkowski [44] and anti-de Sitter [45] superspace.
Higher-spin current multiplets J α(k)α(l) and S α(k) may be realised in terms of the onshell chiral field strengths W α(k) and massless antichiral scalarΦ constrained as These realisations are as follows [45][46][47]: Here W α , W α(2) and W α(3) are the gauge-invariant field strength describing the on-shell vector, gravitino and linearised supergravity multiplets, respectively. The chiral superfields W α(k) for k > 3 are the on-shell gauge-invariant field strengths corresponding to the massless higher-spin gauge multiplets [48,49] (see section 6.9 in [50] for a review). Choosing k = l = 1 in (1.5) gives the supercurrent of the free N = 1 vector multiplet, J αα = W αWα , and the spinor supercurrent of the free N = 2 vector multiplet, S α = W αΦ .
The off-shell gauge-invariant models for massless higher-spin multiplets proposed in [48,49] are not superconformal, although the corresponding on-shell field strengths W α(k) constrained by (1.4a) furnish irreducible representations of the superconformal group, see [50] for the technical details. At the moment we are not aware of explicit realisations of the current multiplets S α(k) for k > 1 in superconformal field theories. Nevertheless, it is of interest to understand the most general structure of three-point correlation functions of such current multiplets, which are compatible with the superconformal symmetry and conservation laws.
In this paper we will restrict ourselves to four-dimensional N = 1 superconformal field theory and to higher-spin current multiplets carrying only undotted or dotted indices, S α(k) and its conjugateSα (k) . We will refer to them as "higher-spin spinor current multiplets". For k = 1 the multiplet S α is just a spinor current multiplet. This case was studied in our previous work [51]. Three-point correlation functions of more general higher-spin current multiplets require additional study and will be explored elsewhere.
The main result of this paper is the most general structure of the three-point function for arbitrary integers k and ℓ. We showed that in all cases eq. (1.6) is fixed by the superconformal symmetry and the conservation equations (1.2) up to a single overall coefficient. However, we found that one has to distinguish two cases depending on values of k and ℓ, see section 5 for details. Additionally, we also studied the mixed correlators where L is the flavour current multiplet and J γγ is the supercurrent.
The paper is organised as follows. A brief review of the two-and three-point building blocks for correlation functions is given in section 2. In section 3 we study a three-point function of higher-spin spinor current multiplets with the flavour current multiplet. We show that this three-point function is fixed by the superconformal symmetry up to two independent real coefficients. In the special case k = 1, our result coincides with that found in our previous work [51]. In section 4 we consider a three-point function of higherspin spinor current multiplets with the supercurrent. In this case the general form of the correlation function is fixed up to three independent real coefficients. In the special case k = 1, our result coincides with that found in [51]. Finally, in section 5 we compute threepoint correlator involving just the conserved higher-spin spinor current multiplets with arbitrary number of dotted and undotted indices. We found that its general structure is fixed up to a single complex coefficient.

Superconformal building blocks
This section contains a concise summary of two and three-point superconformal building blocks in 4D N = 1 superspace, which are important for our subsequent analysis. These superconformal structures were introduced in [14,15], and later generalised to arbitrary N in [16] (see also [19] for a review). A review of the general structure of two-and three-point correlation functions of primary operators is also given. The presentation of this section closely follows [51]. Our 4D notation and conventions are those of [50].

Infinitesimal superconformal transformations
We denote the N = 1 Minkowski superspace by M 4|4 . It is parametrised by the local coordinates z A = (x a , θ α ,θα), where a = 0, 1, · · · 3, α,α = 1, 2. Infinitesimal superconformal transformations are generated by conformal Killing real supervector fields [19,50] As a result, the spinor parameters are expressed in terms of the vector ones The latter satisfy which leads to the standard conformal Killing equation The general solution to eq. (2.5) was given in [50] for N = 1 and in [16] for N > 1. These conformal Killing supervector fields span a Lie superalgebra which is isomorphic to su(2, 2|1). For the purpose of this paper, it suffices to consider the relation The superfield parametersω αβ (z) and σ(z) are expressed in terms of ξ A = (ξ a , ξ α ,ξα) as followsω and can be explicitly found using the components of the conformal Killing supervector, see Refs. [16,19,50] for detail. Due to their action on the covariant derivative (2.7), the zdependent parametersω αβ (z) and σ(z) can be thought of as the parameters of special local Lorentz and scale transformations, respectively. These z-dependent parameters, along with ξ, appear in the superconformal transformation law of a primary tensor superfield, see subsection 2.4.

Two-point structures
Let z 1 and z 2 be two different points in superspace. In 4D superconformal theories, all building blocks for the two-and three-point correlation functions are composed of the two-point structures: where x a ± = x a ± iθσ aθ . In spinor notation, we write Note that (x1 2 )α α (x 21 ) αβ = x1 2 2 δαβ. We sometimes employ matrix-like conventions of [15,19] where the spinor indices are not explicitly written: (2.11b) Since x 2 ≡ x a x a = − 1 2 tr(xx), it follows thatx −1 = −x/x 2 . The notation 'x1 2 ' means that x1 2 is antichiral with respect to z 1 and chiral with respect to z 2 . That is, where D (1)α andD (1)α are the superspace covariant spinor derivatives acting on the point z 1 . Similarly, D (2)α andD (2)α act on the point z 2 . Explicitly, they are given by The superconformal transformation laws of the two-point structures are given by With the aid of (2.14a), it follows that x1 2 2 and I αα (x 21 ) transform covariantly under superconformal transformations: We also note several useful differential identities: Here and throughout, we assume that the space points are not coincident, x 1 = x 2 .
The structures Z i transform as tensors at the point z i , i = 1, 2, 3. For instance, the transformation law of (2.18a) reads We list several properties of Z's which will be useful later (see [15] for details): and hence,X is not an independent variable for it can be expressed in terms of X, Θ,Θ. The variables Z with different labels are related to each other via the identities Eqs. (2.20) and (2.22) imply that and this combination is a superconformal invariant.

Correlation functions of primary superfields
A tensor superfield O A (z) is called primary, if it is characterised by the following infinitesimal superconformal transformation law (2.24) In the above, ξ is the superconformal Killing vector, whileω αβ (z) and σ(z) are the zdependent parameters associated with ξ, see eq. (2.8). The superscript 'A' collectively denotes the undotted and dotted spinor indices on which the Lorentz generators M αβ and Mαβ act. The weights q andq are such that (q +q) is the scale dimension and (q −q) is proportional to the U(1) R-symmetry charge of the superfield O A .
Various primary superfields, including conserved current multiplets, are subject to certain differential constraints. These constraints need to be taken into account when computing correlation functions. It proves beneficial to make use of these conformally covariant operators [15]: from which we can derive these anti-commutation relations One can further prove the following differential identities: In accordance with the general prescription of [14][15][16], the two-point function of a primary superfield O A with its conjugateŌ B is expressed in terms of I, see eq. (2.15): with C O being a normalisation constant.
The three-point function of primary superfields Φ A 1 , Ψ A 2 and Π A 3 has the following general expression [14][15][16]: where the functional form of the tensor H B 1 B 2 A 3 is highly constrained by the superconformal symmetry: (i) It possesses the homogeneity property This condition guarantees that the correlation function has the correct transformation law under the superconformal group. By construction, eq. (2.29) has the correct transformation properties at the points z 1 and z 2 . The above homogeneity property implies that it also transforms correctly at z 3 . The tensor H B 1 B 2 A 3 has dimension (a +ā).
(ii) If any of the superfields Φ, Ψ and Π obey differential equations (such as conservation laws in the case of conserved current multiplets), then H B 1 B 2 A 3 is constrained by certain differential equations too. The latter may be derived using (2.27).
(iii) If any (or all) of the superfields Φ, Ψ and Π coincide, then H B 1 B 2 A 3 obeys certain constraints, as a consequence of the symmetry under permutations of superspace points. As an example, where ǫ(Φ) denotes the Grassmann parity of Φ A . Note that under permutations of any two superspace points, the three-point building blocks transform as The above conditions fix the functional form of H B 1 B 2 A 3 (and, therefore, the three-point function under consideration) up to a few arbitrary constants.
The goal of this paper is to study how N = 1 superconformal symmetry imposes constraints on the general structure of two-and three-point correlation functions involving a higher-spin conserved spinor current multiplet S α(k) and its conjugateSα (k) . Here the complex superfield S α(k) = S α 1 ...α k = S (α 1 ...α k ) is a symmetric rank-k spinor, subject to the following conservation conditions [41] Making use of the transformation law (2.24), it can be shown that eqs. (2.33) are consistent with superconformal invariance provided S α(k) is a primary superfield with weight (q,q) = (1 + k 2 , 1) and dimension 2 + k 2 . Let us first consider the two-point correlation function of S α(k) and its conjugateSα (k) . Adapting the general prescription (2.28) to this case leads to where A is a real coefficient. Using the identities (2.17), along with one may verify that the correlator (2.34) respects the conservation conditions at non-coincident points z 1 = z 2 .
In the case of k = 1, the expression (2.34) is in agreement with [15,19]. Various threepoint functions involving S α and its conjugateSα have been studied in detail in [51].
In our recent work [51], we found that the three-point correlator with two spinor current insertions and a flavour current multiplet has two linearly independent functional structures with real coefficients c 1 and d 1 : where Here we adopt the notation in which X k ≡ (X 2 ) k/2 .
Since H β(k)β(k) is Grassmann even, the most general expansion we can write is (3.10) Constraints (3.9b) and (3.9d) immediately tell us that which leaves us with β(k)β(k) . Since A β(k)β(k) and B β(k)β(k),γγ can be constructed using only tensors X αα , ε αβ , εαβ, it is not hard to list all possible independent structures consistent with the homogeneity property (3.5) by performing irreducible decompositions into symmetric and antisymmetric parts in dotted and undotted indiices: Here a 1 , b 1 , b 2 are arbitrary complex coefficients. To continue it is convenient to introduce auxiliary commuting complex variables (u α ,wα), with the property u α u α =wαwα = 0. Given a tensor superfield T α(k)α(l) , we can associate to it the following index-free superfield T (k,l) (u,w) := u α 1 . . . u α kwα1 . . .wα l T α 1 ...α kα1 ...α l , (3.13) which is homogeneous of degree (k, l) in the variables u α ,wα. Making use of these auxiliary variables and their corresponding partial derivatives (∂/∂u α , ∂/∂wα) allows us to also convert the conformally covariant derivatives into index-free operators, for example: 14) These nilpotent operators decrease the degree of homogeneity in u α andwα.
Using the notation introduced, eq. (3.12) turns into where we have defined (P · X) := 2Θ αΘα X αα , (P · K) := 2Θ αΘα K αα . (3.18) Throughout the paper, we will also often employ the following shorthand notation The differential constraints (3.9a) and (3.9c) now read A useful observation is that This means that a 1 is an independent coefficient, since the first term in (3.16) already satisfies conservation equations. It can be explicitly checked that In deriving (3.23) and (3.24), the following easily verified identities have been used: Xα α X 2p+2 ,Qα At this stage, we are left with two unconstrained complex parameters, a 1 and b 1 . The next task is to check if the flavour current conservation equations (3.8c) are satisfied. Checking conservation laws on z 3 requires more work as there are no identities that would allow differential operators acting on the z 3 dependence to pass through the prefactor of (3.4). Following the procedure carried out in [51], let us first express (3.4) as Here the I-operators are the higher-spin extensions of (2.15). Specifically, we define (3.28) Its inverse will be denoted bȳ (3.29) These operators satisfy It should be kept in mind that the spinor indices can be raised or lowered in accordance with (2.10b). This allows us to write By rearranging the operators in the three-point function and taking into account their Grassmann parities, one may express the left-hand side of (3.27) as for some function H˙γ (k)α (k) (X 1 , Θ 1 ,Θ 1 ). Comparing eqs. (3.27) and (3.33) allows us to relate H and H where we have used the identitȳ In order to compute the expression (3.34), we further note that Direct computation yields , (3.37) which can again be written in index-free notation as with (v ·w) = εαβvαwβ. We now observe that As a consequence of (2.27), the conservation conditions on the flavour current multiplet are equivalent to It is straightforward to show that (3.38) indeed satisfies (3.40) for arbitrary a 1 and b 1 .
The final step is to impose the reality constraint, (3.7), which requires where A and B are real. Thus, the higher-spin correlator (3.4) is fixed by the N = 1 superconformal symmetry up to two independent, real coefficients A and B.
As a result, the final form of (3.4) proves to be where the tensor H β 1 ...β kβ1 ...β k X 3 , Θ 3 ,Θ 3 has the compact expression Setting k = 1, direct comparison with (3.3) shows that we have an agreement provided the parameters are related in the following way: In a more general setting, one may also consider 44) in which case the only difference with (3.42a) is that the superfieldS ′α (k) is not the complex conjugate of S α(k) , i.e. (S α(k) ) * =S ′α (k) . It still obeys the conservation conditions Dα 1S ′α 1 ...α k = D 2S′α (k) = 0. Since the reality constraint (3.7) no longer holds, the correlator (3.44) is determined up to two complex coefficients, a 1 and b 1 .
The three-point function of the type Sα (z 1 )S α (z 2 )J γγ (z 3 ) was constructed in [51]. It is determined up to three independent, real coefficients: with Hβ β γγ (X, Θ,Θ) = (4.3b) Let us consider Sα (k) (z 1 )S α(k) (z 2 )J γγ (z 3 ) . Using the prescription (2.29), the correlator takes the form Following similar analysis as in section 3, the tensor H β(k)β(k), γγ is required to satisfy these properties: • Homogeneity: H β(k)β(k), γγ (λλX, λΘ,λΘ) = λ −(k+1)λ−(k+1) H β(k)β(k), γγ (X, Θ,Θ) , (4.5) • Reality: • Conservation conditions: Furthermore, the conservation equations of the supercurrent, eq. (4.1), demand that the following be satisfied: First of all, the fact that H β(k)β(k),γγ is Grassmann even and obeys the differential constraintsD 2 H β(k)β(k),γγ = 0 and Q 2 H β(k)β(k),γγ = 0, imply that it has the general form H β(k)β(k),γγ (X, Θ,Θ) = A β(k)β(k),γγ (X) + B β(k)β(k),γγ,δδ (X)Θ δΘδ . (4.9a) We then write all possible independent structures consistent with (4.5). This is done by decomposing A β(k)β(k),γγ and B β(k)β(k),γγ,δδ in terms of their irreducible components where a 1 , a 2 and b 1 , . . . b 6 are arbitrary complex coefficients. Before imposing conservation conditions (4.7), it is useful to contract the indices of H β(k)β(k),γγ with auxiliary variables. We thus introduce H (k,k),γγ (X, Θ,Θ; u,w) := u β 1 . . . u β kwβ1 . . .wβ k H β(k)β(k),γγ where K γγ is defined the same way as in (3.17), K γγ = u αwα X αγ X γα . In this basis, the (tilde) coefficients are related to those in (4.9c) by the rulẽ As in the previous section, the first order constraints in (4.7) are equivalent to Let us first look at the condition D (−1,0) H (k,k), γγ = 0. With the help of identities (3.26) and considering only the terms linear inΘα, we find the following constraints on the coefficients After some lengthy calculations, it can be shown that the terms cubic in the Grassmann variables (that is, In a similar way, one can computeQ (0,−1) H (k,k), γγ and show that the terms linear in Θ lead to the requirements while the terms proportional to Θ 2Θδ give The system of equations (4.13)-(4.16) turn out to be consistent and can be solved in terms of three independent coefficients which we choose to beã 2 ,b 1 andb 3 : Keeping in mind the relations (4.11), one may rewrite (4.17) in the basis (4.9). Choosing the three independent coefficients to be b 1 , b 2 , b 4 , we find that It remains to impose conservation equations on the supercurrent J γγ , eq. (4.8). As in section 3, this can be done by appropriately rearranging the operators in the correlator and transforming the tensor H β(k)β(k), γγ . Let us rewrite (4.4) in terms of the I-operators: On the other hand, the same correlator can be written as 20) Thus, knowing H, the tensorH can be computed using the formula where the identity (3.35a) has been used.
We now substitute the explicit form of H β(k)β(k), γγ into (4.22). This is given by (4.9), but with b 5 = b 6 = 0, in accordance with the constraints (4.18). After some tedious calculations and making use of (3.36), along with the identities we obtain the following expression (4.23) We recall that b 1 , b 2 and b 4 are independent. We have also defined V αα := X αβvαwβ .
Going back to eq. (4.20) and relabelling superspace points, we see that By virtue of eq. (2.27), the conservation law of J γγ requires that the following equations must hold: It is not hard to verify that (4.23) satisfies (4.25) for an arbitrary choice of complex The last step is to impose the reality condition (4.6): HereH β(k)β(k),γγ (X, Θ,Θ) means that we are taking the complex conjugate of the expression (4.9). This implies that where A, B, C are real coefficients.
• Conservation conditions: However, these structures prove to be linearly dependent of the others. More precisely, one can prove that (v ·w) l (ΘXw)(uXΘ) X k−1 (1,1) Imposing conservation laws thus leaves us with only two independent coefficients, which we choose to be a 1 and b 1 .