Towards general super Casimir equations for $4D$ ${\mathcal N}=1$ SCFTs

Applying the Casimir operator to four-point functions in CFTs allows us to find the conformal blocks for any external operators. In this work, we initiate the program to find the superconformal blocks, using the super Casimir operator, for $4D$ ${\mathcal N}=1$ SCFTs. We begin by finding the most general four-point function with zero $U(1)_R$-charge, including all the possible nilpotent structures allowed by the superconformal algebra. We then study particular cases where some of the operators satisfy shortening conditions. Finally, we obtain the super Casimir equations for four point-functions which contain a chiral and an anti-chiral field. We solve the super Casimir equations by writing the superconformal blocks as a sum of several conformal blocks.


Introduction
In a general CFTs, it is possible to write the product of two primary operators as a sum of conformal families, given by primary operators and their descendants. This information is encoded in the OPE expansion. The only unknown variable in the OPE is the three-point function coefficient. The fact that conformal symmetry completely fixes the three-point function up to a constant is possible thanks to the absence of conformal invariant structures for any three-point function. Although it is natural to expect a similar statement when supersymmetry enters the game, this is not true in general. The existence of superconformal invariant structures in a generic three-point function was established long ago in the case of four dimensional N = 1 SCFTs [1,2]. Those structures prevent, in general, to write the product of two superconformal primaries as a sum of superconformal families 1 . Take, for example, the three-point function of a general scalar multiplet O 1 , with superconformal charges q andq, with its conjugated and another scalar long multipletÕ (ℓ) in an N = 1 SCFT. Superconformal symmetry fixes the three-point function up to four unrelated constants: see (4.57) and (4.58).
Although supersymmetry completely fixes the relation between the superprimary operator of O and its super-descendants, the fact that the λ i coefficients are unrelated implies that there are four distinct conformal families appearing the sOPE of O × O † instead of one single superconformal family. If instead of a long multiplet O we study a short-one, for example a chiral multiplet, the shortening conditions will imply relations among the λ i coefficients [4][5][6][7], but even in those cases it is not always possible to fix all the coefficients, see for example [3,[8][9][10][11][12].
Furthermore, if one wishes to study the λ 2 , λ 3 and λ 4 coefficients in (1.1) with, for example, bootstrap techniques [13], one has to start with a four-point function which includes all the superconformal invariant terms, otherwise, it would not be possible to perform a complete conformal partial wave expansion. The proliferation of such superconformal invariants for general N-point function in SCFTs has prevented the study of such four-point functions. As we will see, a four dimensional N = 1 four-point function of scalars with vanishing total U(1) R -charge has 35 nihilpotent structures. Therefore, one has a total of 36, and not just one, independent functions of the conformal invariant cross ratios. This is different from the non-supersymmetric case, where there is only one function of the conformal invariant cross ratios in the four-point function of scalar operators, and it is, as we will see, more closely related to the case where there are spinning correlators [14].
The aim of this article is to initiate the program for constructing the superconformal partial waves for general four dimensional N = 1 SCFTs using the super Casimir operator. We begin this program by working out the most general four-point function for scalar with total zero U(1) R -charge. Since such four-point function consists of 36 functions of the supersymetryc cross-ratios in the most general case, we look at four-point functions where at least two of the external operators correspond to short multiplets, in this case, a chiral superfield, Φ, and a current multiplet, J . Imposing the shortening conditions, we find that the four-point functions with a chiral and an anti-chiral only depend on three independent functions. Acting with the super Casimir operator on such four-point functions we find a system of coupled eigenvalue equations in the s-channel, while in the t-channel we also find a constraint. In order to solve the super Casimir equations we study the three-point functions, which tell us which operators are being exchange between the lowest components and the first Q(Q) descendants. This allows us to give an ansatz which solves the super Casimir equations. When studying the schannel of the OO † ΦΦ correlator, unlike the non-supersymmetric case, the solutions to the super Casimir equations tell us that one of three-point function coefficients in (1.1), λ 4 , is zero.
This article is structured as follows. In section 2 we find all the four-point function superconformal invariants, which consists on two supersymmetric cross-ratios and 35 nilpotent invariants. This allows us to write down the most general four-point function with zero U(1) R -charge in (2.60). In section 3 we study the cases where some of the operators correspond to a chiral or a current multiplet. The shortening conditions coming from such multiplets imposes several constraints to the four-point functions. In section 4 we apply the super Casimir operator on four-point functions where one of the external operators is a chiral and another an anti-chiral and we solve the super Casimir equations by decomposing the superconformal blocks as a sum of conformal blocks. We finish in section 5 with discussions and outline future lines of research.

Constructing the Four-Point Function
The most general four-point function with zero U(1) R -charge is given by These coordinates satisfyDαz + = D α z − = 0, where the superderivatives are given by (2.5) The rule for raising and lowering spinor indices is, as usual, Finally, we can write any four-vector as a 2 × 2 matrix with the help of the Pauli matrices, and now the chiral coordinates read The supersymmetric version of the distance of two points, These coordinates transform under the superconformal group as, beingω,ω, σ,σ, τ andτ the parameters of the infinitesimal superconformal transformations. Note that the parameters have the following constraintŝ It is also useful to define where ī j = x 2 i j . For N ≥ 3 points, z 1 , z 2 , · · · z N it is convenient to define the supersymmetric coordinate Z = X, Θ,Θ , where the four-vectors X and its conjugatedX are given by X r(s t) = x rsxs t x tr s r r t ,X r(s t) = −X r(t s) , which transform homogeneously at z r , as can be checked using (2.10): As mentioned in the introduction, unlike the non-supersymmetric case, we can construct a superconformal invariant for the three point function . (2.18) As expected, (2.17) vanishes in the bosonic limit. This superconformal invariant was first written by Park [1] with a different notation (2.20) As we will see in the next section, Q and Q 2 are the only spinless superconformal invariant that can be written in the three-point function.

Four-Point Function Invariants
In flat space we can always go from point i to point j by first passing through a third point k: x i j = x i k + x k j . A similar statement holds for the supersymmetric version of the distance x ij , (2.21) We can further extend this notion to the vectors given in (2.14) X r(s t) = X r(s u) + X r(u t) − 4 iΘ r(s t)Θr(t u) , (2.22) and the Gassmann coordinates (2.15), Θ r(s t) = Θ r(s u) + Θ r(u t) ,Θ r(s t) =Θ r(s u) +Θ r(u t) . (2.23) At first sight, one might think that there are six four-vectors that transform homogeneously at a given point, say z 1 . But, from (2.22) and using that X r(s s) = 0, we end up with only two at every point. The same is true for the Grassmann coordinates. In general N-point functions, there will be N −2 vectors and Grassmann coordinates [1,36]. For the moment, we will work on the basis r(s t) = {1(2 3), 1(4 3)}. Later, we will see that every superconformal invariant can be written in this basis. This also shows that Q and Q 2 , as given in (2.17) and below, are the only three-point function invariants.
In non-supersymmetric theories, there are two conformal invariant functions The supersymmetric extension of these cross-ratios are given by (2.25) Since their θ i → 0 limit is non-vanishing, every other nilpotent superconformal invaritant that we write will be multiplied by an arbitrary function of these u and v terms.
Using (2.21) we can relate any u r s,t u and v r s,t u with any other pair u r ′ s ′ ,t ′ u ′ and v r ′ s ′ ,t ′ u ′ plus some (θθ) term. The simplest way to do so is by using the X and Θ coordinates. In terms of the Xs we have where we have defined, One might think that terms such as are new superconformal invariants, but from (2.14) and (2.22) it is easy to see that and a similar relation holds for tr X a ·X b /X 2 b . If instead of X 1(a 3) , we take X i(a j) for i = 2, 3, 4 we find, Terms such such as X 2 i(··· ) /X 2 j(··· ) are superconformal invariants only if i = j. A trace with more than two X terms, tr (X 2n ) can always be rewritten as a sum of the cross-ratios by using since we only have two X-terms that can appear in such expression.
We can now easily relate u ab,ij with u ij,ab where · · · means terms proportional to θθ. A similar relations hold for the v's defined in (2.31). Finally, note that, We can perform a similar exercise for any other combination. This shows that any arbitrary function f (u ab,cd , v ij,kl ) can be written as f (u 24,31 , v 42,13 ) up to fermionic terms, as already mentioned.
Therefore, our first term of the Θ-expansion of F (z 1 , z 2 , z 3 , z 4 ) is just an arbitrary function of the cross-ratios, as expected, At the first Θ level, the only scalar we can construct with zero U(1) r -charge are for i = 1, 2, 3, 4. Using (2.32), we can write any of those terms as a function of times an arbitrary function of the cross-ratios. We now show that we can write any term in (2.37) as a function of, times an arbitrary function of the cross-ratios up to O Θ 2Θ2 . In order to do so, we need to generalize the relation (2.52) and (2.53) in [2]: (2.39a) which can be easily verified using For example, where the dots means higher powers in ΘΘ. We have repeatedly used (2.14), (2.15), (2.22), (2.23) and (2.29). As discussed before, for the 3(i 1) basis we only need i = 2, 4. Therefore, we can have either a = b, b = c or a = c in (2.43). Using (2.32) we find that, where (a b c) was defined in (2.38). This shows that we can always change the basis by using (2.14) and (2.15). For example, we can write any X 2(a b) in terms of X 2(1 3) and X 2(1 4) , and since we can change our original basis from X 1(a 3) to X 1(b 2) we can have a result similar to (2.44). Thus, the (a b c) terms in (2.38) are a basis for all the possible superconformal invariant at the first ΘΘ level.
Are all the eight different (a b c) terms independents? In order to answer this, we perform a suitable superconformal transformation and set z A 1 = (0, 0, 0) and z A 3 = (∞, 0, 0). In this basis, we find that, which indeed shows that the eight (a b c) terms are linearly independent. Thus, our F (z 1 , z 2 , z 3 , z 4 ) function now reads At the Θ 2Θ2 -level, we can write the following superconformal invariants (2.47a) Using the Fierz identities, we can relate any of these superconformal invariants with (a b c) (d e f ). First, we can check that for any two vector x and y, which we can choose as X 1(2 3) and X 1(4 3) . Thus, these terms reduce to the (a b c) (d e f ) type of terms.
Note that we can write any Θ a X bXcΘd as a function of Θ aΘd and Θ a X 2X4Θd using (2.32), and similarly for theΘ situation. Now we can check that The Θ a X bXcΘd Θ eXf X gΘh terms are trivially reduced to the previous cases except for a = e = 2 and d = h = 4. In those cases, one can finally show that (2.51i) We can write any of the remaining 45 terms as a linear combination of (2.51). This can be verified by with the help of the following relations (a m b) (a n b) From what we learned from the previous case, we can rotate our basis and reach any {i (j k)} basis. Thus, we have exhausted all the possibilities at order Θ 2Θ2 . Therefore, our f function is, where the C a i are arbitrary function of the cross-ratios and non-zero only for the cases listed in (2.51).
The Θ 3Θ3 case is now almost straightforward. From the previous discussion, we know that we can only have structures of the type The only question is, which structures are independent. Since our basis 1 (a 3) can only take two values, a = 1, 2, we have only two types of terms Therefore, since the only terms that we can have are which can be arranged as, Finally, now it is easy to see that the Θ 4Θ4 term is unique, and higher order in ΘΘ vanish. Thus, the most general solution for F (z 1 , z 2 , z 3 , z 4 ) = F (Z a ) is given by The form of (2.60) allows to change X 1(23) by X 1 (43) in (2.1), and this will only imply a shift in the f (u, v) functions. Furthermore, we can also make a change of basis, as we will later do, by following the same procedure as in (2.44). This will allow us to impose constrains at any point of the four point function.

Imposing Constraints to the Four-Point Function
Plugging (2.60) into (2.1) give us the most general four-point function, but, as we already know from experience with the three-point functions, shortening conditions impose relations among the different structures. In this section we will study such constraints in the case of the four-point function. We will study three systems of increasing complexity. The simplest system consist only on chiral and anti-chiral fields We are also interested in a mixed system of chirals and current-multiplets, where the current-multiplets satisfy Finally, we will study a system with only conserved currents

From Superderivatives to Superconformal Covariantderivatives
In order to understand the constraints (3.2) and (3.4) we first need to see how the super-derivatives acts on Z a(bc) coordinates. It is not hard to generalize the action of the super-derivatives on the Z a(bc) given in [2]: By inspecting (3.6) it is natural to define the superconformal covariant derivatives 2 . (3.7b) As we will see later, the shortening conditions (3.2) and (3.4) acting on the fourpoint function take a simpler form as a superconformal covariant derivative (3.7) acting on F (Z a ).

ΦΦ ΦΦ
The simplest four-point function that we can solve is where F (Z a ) is given by (2.60).
The most straightforward way to solve (3.8) is by using a system which is explicitly chiral at z 1,3 and anti-chiral at z 2,4 . A basis with such behavior is Although the cross-ratios U and V , have the right chirality for every z i , they do not reduce to the standard cross-ratios in the bosonic case. Thus, we will use which in the bosonic limit are as desired.
Using (3.6), we can write the constraints to the four-point function as differential equations acting on the F function in (3.8). Explicitly, It is simple to see that the first two constraints totally fix F (Z a ) setting all the Bs, Cs and E 1 functions to vanish. Therefore, with u and v given above.
Since we never used the chirality property of the Φs, one might wonder what happens when the chiral fields are replaced by arbitrary scalar multiplets. Taking the scalar fields O 1 and O 2 with U(1) R -charges (q 1 ,q 1 ) and (q 2 ,q 2 ) respectively, we find with q 1 + q 2 =q 1 +q 2 + 2q. Note that the fact that (3.14) and (3.15) do not depend on any nilpotent structure simplified the computation of the super Casimir acting on the four-point function [27,34].

J J ΦΦ
More interesting is the case where the first two fields are replaced by two currentmultiplets, Again, we first set, as in the previous computation, a basis which is explicitly chiral at z 3 and anti-chiral at z 4 , which as before is given by We define the cross-ratios as in (3.11).
A short computation shows with D andD defined as in (3.7). The constraints at z 3 and z 4 are the same as in (3.13). Thus, the constraints at points 2, 3 and 4 take a simple form: Note that we haven't written the constraints coming from the conservation of J (z 1 ). In order to find a simple set of equations for those constraints, we will rotate the basis for the Zs in F (Z a ).
The simplest constraint to solve is (3.20a), which implies 6 and C 17 , which remain unconstrained. Chirality at z 3 , (3.20b), relates the B terms and sets to zero the remaining C terms. (3.20d) does not impose any new constraints, while (3.20c), give us one differential equation In order to apply the constraints coming from the conservation of J (z 1 ), we rewrite our coordinates and the correlator. We first note that 25) and also .
Thus, now we can write our correlator as, . (3.28) The constraints coming from J (z 1 ) now read It is not hard to check that (3.29b) does not not give any new equation, and (3.29a) just gives us (3.24) again. Therefore, finally have the four-pint function with (3.24) relating the terms.
As seen above, the chirality constraints (3.20a,3.20b) are very restrictive. Indeed, we can replace the currents for general operators O 1 and O 2 such that the total U(1) R -charge is zero, i.e., q 1 + q 2 =q 1 +q 2 It is easy to see that (3.30) and (3.31) corresponds to the s-channel four-point function. In order to bootstrap this system, we also need the t-channel propagator. In the Bosonic limit, this propagator is given by In order to write the full supersymmetric propagator, we first need to define a basis. In this case, we will work with Z 4(a2) . In this basis, the cross-ratios which have the right bosonic limit are given by The right four-point function should have the right 1 4 4 1 factor and the right (anti-)chirality for (z 4 )z 3 . Thus, the full supersymmetric four point function in the tchannel is given by with F (Z a ) given by (2.60) with 2 → 1, 4 → 3. The constraints now read where again, we will compute the constraints at z 4 after rotating the basis. Just as before, (3.36a) set to zero all the B i , C i , D i and E 1 , except for B 1 , B 2 , B 5 , B 6 , C 1 , C 5 , C 6 and C 17 . (3.36b) set to zero the remaining C i , while imposing B 5 = −B 1 and B 6 = −B 2 . No new constraints come from (3.36c), while (3.36d) impose a differential equation, Finally, imposing the constraints at z 4 will not give us any new constraint.
Our final four-point function is with (3.37) relating the functions of the cross-ratios.
If we replace the currents for long-multiplets, the four-point function is 3.4 J J J J For a system solely composed of current multiplets J , which are not necessarily the same current, we choose our basis to be Z 1(a3) , with the cross-ratios given by (3.11). Naively, one would write the four-pint function as which has the right s-channel limit. But the relations (3.18) and (3.19) tell us that we should have only ī 1 1 i terms. Thus, the supersymmetric four-point function has to be with F (Z a ) given by (2.60). We can check that (3.41) has the right conformal weights, and it also reduces in the bosonic limit to the usual s-channel four-point function.
Imposing D 4 2 J (z 4 ) = 0 impliesD 2 4 F (Z a ) = 0, which implies The conservation equation at z 2 ,D 2 2 J (z 2 ) = 0 now reads (Q 2 + Q 4 ) 2 F (Z a ) = 0, which imposes two relations among the remaining C i terms, In order to impose the constraints at z 1 , we need to rotate our basis. Following the same procedure as above, we rotate to Z 3(a1) . The four-point function now reads which can be written as Imposing D 1 2 J (z 1 ) = 0 implies (Q 2 + Q 4 ) 2 F (Z a ) = 0 in the new basis, which give us two more relations The remaining constraint give us 16 long differential equations, see below. Finally, the four-point function is given by and We also have several differential equations, for example plus several other differential equations involving derivatives, which can be find in the attached Mathematica file. The equation involving the C i terms are the supersymmetric generalization of the ones found in [22].

Casimir
Having the full supersymmetric four-point functions, we are ready to apply the super Casimir operator to them. First we write the N = 1 superconformal algebra following the conventions of [3]. 3 The bosonic part of the algebra is given by while the anti-commutators are finally, the remaining of the algebra is given by and the rest of the (anti-)commutators vanish. We can represent the operators in the algebra with differential operators acting on the coordinates. Following [3], where q = 1 2 ∆ + 3 2 r ,q = 1 2 ∆ − 3 2 r , with ∆ being the scaling dimension and r the U(1) R -charge of the operator where the generators act upon, s ab is the proper rotation matrix and s αβ and sαβ are the proper projections.
The quadratic super Casimir for this algebra is which for our generators reads When acting on two different points, (4.6) acts as a differential operator (see (4.4).) Explicitly, when acting on two scalar supermultiplets, (4.6) reads Although it would be nice to write (4.8) in term of derivatives acting on Z (see (3.7)) we found that this is not as systematic nor straightforward as one might expect. Instead, it is simpler to use the superconformal transformations and send one coordinate to z j = (∞, 0, 0). In this frame, the pair of Z a coordinates is easily written as a pair of z ij coordinates.
As mentioned above, we will study the simplest four-point functions with non-zero nilpotent invariants, which are (3.30), (3.31), (3.38) and (3.39). Before we proceed, it is instructive to study the ΦΦ ΦΦ propagator. We already know that the propagator is given by (3.14) Expanding A 1 as a sum of superconformal blocks, now it reads When acting with the quadratic super Casimir (4.8) on it, we find the eigenvalue equation 4.10) with λ given in (4.7). This equation was already found in [34] using the super-embedding formalism. 4 In order to solve this equation, we look at the three-point function [5]. The only solution for the three-point function consistent with the constraints is where X ℓ denotes symmetric-traceless and ∆ is the dimension of O. Looking at (3.14), we see that A 1 is proportional to the φφ * φφ * four-point function. (4.11) tell us which components of the O (ℓ) supermultiplet appear in the φφ * OPE: With this information, we can make an ansatz for G 0 0,O : where g ∆,ℓ (u, v) are the conformal blocks [15,16] g α,β (u, v) = g 0,0 α,β (u, v) , (4.14) and with u = zz and v = (1 − z)(1 −z).
Plugging this ansatz into (4.10), we can fix the c i coefficients: which agrees with the results found originally in [5]. We stress that in (4.13), the cross-ratios are the supersymmetric cross ratios and not just the bosonic limit.

J J ΦΦ s-channel
More interesting is the case of mixed operators. We begin studying As mentioned before, this propagator corresponds to the supersymmetric version of the s-channel.
In order to solve the super Casimir equations, we first expand our four-point functions as a sum of superconformal blocks Applying the super Casimir (4.8), 5 we find three eigenvalue equations: Before we proceed, we point out that Casimir equations where different structures mix were first found when studying seed blocks [21]. Unlike them, we do not find a nice "nearest-neighbor interaction" interpretation. The only differential operators in the G Just as we did in the previous section, in order to solve these equations, we will look at the three-point functions of various descendants. The superfields J , Φ and an external long multiplet O (p,p) can be expanded as (4.20d) The contribution for the G 1 1,O and G 1 2,O superblocks can be read from the Jj α φψα correlator: 6  Since the only operator that can be exchanged is an O (ℓ) ∆ , see (4.11), the only threepoint function for the current multiplet that we need to study is [3,8,9] and X + = 1 2 X 3(12) +X 3 (12) , X − = i X 3(12) −X 3 (12) . (4.27) Note that under the exchange z 1 ↔ z 2 , t (ℓ) ± . For simplicity, we will only consider the case where the current multiplets are the same. Thus, for even spin, only t + contributes, while for odd spin only t − contributes.
Looking at (4.25), we know that the OPEs between the lowest components of J are (4.28b) 6 We can also read the contribution from other correlators, any choice will be related by supersymmetry. For example, one could also use jα Jψ α φ * = 4 1 We can now make an ansatz for A 1 . Looking at (4.12) and (4.28) we have What about the G 1 1,O and G 1 2,O functions? From (4.23) we know that the first Q descendant of J and Φ will give us an ansatz for them. From (4.12), we can read those OPE A similar expression can be obtained from (4.28) for both even and odd spin.
The exchange of an operator in the {∆, (ℓ + p, ℓ)} representation of the conformal group will generate a W seed ∆,ℓ,p seed block, while an operator in the {∆, (ℓ, ℓ + p)} representation, which has the same Casimir eigenvalue, will generate the corresponding W dual seed ∆,ℓ,p dual seed block [21] (see also [23]). Thus, the OPEs (4.30) and (4.31) together with (4.23) tell us 7 We have introduced an arbitrary normalization constant a 0 (which for our case is a 0 = i), since it is possible to find different normalizations for the Q descendant of the supermultiples in the literature. This constant does not affect our results.
Plugging (4.29a) and (4.32) in (4.19), we find If we plug the ansatz for odd spin, (4.29b), instead of the even spin ansatz, we find The c 1 and c 2 terms agree with [31].
A few comment are in order before we proceed. First, we set the norm of the first block always as one. The super Casimir equations are independent of such normalization. In order to properly fix it, one can, for example, use the techniques from [9]. Second, in order to find either (4.34) or (4.35), we only needed to solve the eigen equations for G 1 0,O and G 1 1,O . The equation for G 1 2,O was automatically satisfied. 8 Third, as we mentioned before, the blocks coming from the lowest component of the multiplets are independent of the a 0 normalization.

t-channel
Now we turn our attention to the t-channel superblocks. We already computed the four-point function in the previous section, We expand this correlator as a sum of blocks As before, we apply (4.8) to (3.38), and we find three eigen equations The three-point function between J and Φ is given by (21) . (4.38) Imposing (3.2) and (3.4) 9 , we find three possible operators that can be exchanged in the t-channel: Note that in (4.39a), the dimension of the exchanged operator is ∆ = q − ℓ, which for ℓ = 0 is a known short multiplet O =Φ. Regardless of this, we will solve the super Casimir for an operator with arbitrary dimension.
Just as before, we will look at the OPEs between φ × J, ψ α × J and φ × j α and from there make an ansatz for the A 1 , B 1 and B 2 functions. Taking the term proportional to (θ 2θ4 ) in the LHS of (3.38), we find where 10 For (4.39a), we find the following OPEs For (4.39b), we find the following OPEs Finally, for (4.39c), we find the following OPEs In all three cases, ψ α × J ∼ j α × φ. The reader should keep in mind that although A and its QQ descendants have the same U(1) R -charge as the supermultiplet in (4.42), this is no longer true in (4.43) and (4.44).
Thus, in the case the exchange operator is O (ℓ,ℓ) (q−ℓ/2,−ℓ/2) , we have and 4a 0 G 2 Plugging this ansatz in (4.37) gives us , (4.48a) , (4.48b) , (4.48c)  Plugging this ansatz in (4.37) gives us , (4.51d) Finally, for the exchange of an O (ℓ,ℓ+1) ((−3+2∆−2q)/4,(3+2∆−2q)/4) operator, we have G 2 0,O = g 2−q,q−2 ∆+1/2,ℓ + c 1 g 2−q,q−2 ∆+3/2,ℓ+1 , (4.52) and 4a 0 G 2 (4.53) Plugging this ansatz in (4.37) gives us (4.54f) We point out that the superconformal blocks for the lowest components agree with the results found in [31]. Just as before, the normalization a 0 is irrelevant for the blocks of the lowest components, as expected. Again, we only needed to solve two eigenvalue equations and the third was identically satisfied. In the case of (4.42) we could have used either G 2 1,O or G 2 2,O . In the case were the lowest component of the exchanged long multiplet was not present in the OPE, things were different: for (4.43), the eigenequation for G 2 0,O we were able to fix all the c i coefficients but one, which was fixed using G 2 2,O , while G 2 1,O did not give us any new information; the converse statement is true for (4.44), where we used G 2 1,O to fix the remaining constant, while G 2 2,O was identically satisfied.

O O ΦΦ
So far, we have studied a special case where we only have short multiplets. As we saw in the previous section, we can replace the current multiplet and still have a simple fourpoint function. In (3.31) and (3.39) we replaced the current multiplet for two arbitrary long multiplets such that the total U(1) R -charge is zero. We will now work in the special case where q 1 =q 2 and q 2 =q 1 . We will assume q 1 =q 1 . The special case where the long multiplet has zero U(1) R -charge is very similar to the case of two current multiplets and it was studied in detail in [31]. Since the lowest component of the long multiplet is a complex field, we cannot make a distinction between even and odd spin cases, as in (4.29). Therefore, the ansatz for the A 1 function will be very similar to the one given for the chiral case, see (4.13). In the t-channel, we will study the same long multiplets being exchanged as in (4.39) and we will obtain very similar results to (4.48), (4.51) and (4.54).

s-channel
In the case where we replace the current multiplets by a long and its conjugated, (3.31) now reads First, we expand the correlator Acting with super Casimir (4.8) on the chirals, unsurprisingly, we find the same eigen equations as in (4.19).
Again, we look at the three-point functions in order to make an ansatz for A 1 and the B i functions. We know that the only contribution to the super blocks comes from a O (ℓ) (∆/2,∆/2) multiplet (4.11). The three-point function between the longs and such operator was already written in the introduction, The subscript t ℓ i± means that under z 1 ↔ z 2 , t ℓ i± → ±(−) ℓ t ℓ i± . The structures appearing in (4.57) are the same as the ones appearing in (4.25), but since there are no constraints that can be imposed to the long multiplets, there is no equation relating them.
The m p+p × m * p+p OPE, being m p+p the lowest component of the long multiplets, is of the same form as the φ × φ * OPE, (4.12). Thus, the ansatz for A 1 is given by (4.13).
For the B i functions, we again look at the term proportional to θ 2θ4 . This turns out to be again (4.23), replacing the dimension fo the current multiplet by p +p. The m p+p × n † (p+p)α OPE, being n (p+p)α the firstQ descendant of the long multiplet, is of the same form as the J × j α OPE, (4.31). Thus, our ansatz for the Bs is given by (4.32).
Summarizing, our ansatz for G 3 0,O is Solving the eigenvalue equations with these ansatz, we obtain , (4.62a) This result shows us that the super Casimir equation (4.8) fixes a coefficient of the three-point function! If we look at the contribution coming from t (ℓ) +4 in (4.58), we see that the only operator which is being exchanged between the superprimaries is the D in (4.12). Setting coefficients c 1 , c 2 and c 3 to zero in (4.59) while giving the most general form for the other G 3 i,O superblocks, (4.60), and solving the super Casimir equations, tell us that the only possible solution is c 4 = 0, which implies λ (4) + = 0. Looking only at the superconformal primaries, one would have never been able to fix this coefficient [10]. This is a very striking and unexpected result. As we will see later, acting on the t-channel, the super Casimir will also fix the three-point function coefficients for a large family of long multiplets.
It is also worth noting that the super Casimir equations do not fix the coefficients c 1 , c 2 and c 3 . This is also to be expected: from the three-point function we see three different three-point function coefficients which will enter in the c 1 , c 2 and c 3 coefficients, see [10], leaving, effectively, three unconstrained coefficients.
and one constraint 12 It is easy to see that there is no constraint for the current multiplet case, since there p = 1. Although the constraint seems surprising, it is easy to understand. First we note that (4.66) is proportional to All these structures have the property to be chiral at z 2 . Therefore, we see that the super Casimir operator mixes all the possible structures allowed by the symmetries of the operators on which it acts on, just like in the seed blocks case. The difference now is that shortening conditions coming from the remaining operators in the four-point functions are invisible to the super Casimir. Those shortening conditions also impose relations between the different terms in the four-point function, unlike the seed block case, where the structures are fixed and there is no condition which will kill their contribution, therefore, there will never be a constraint.
The next step can be guessed by the reader: we will look at the three-point functions and see which operators are being exchanged. Most of the work was already done in (4.39), where we note that now (4.39a) does not have a fixed power in the denominator, and since there is no conservation equation for the long multiplet, there is another structure. The three-point function is (21) . (4.68) The long multiplet version of (4.39) reads ((∆−p+p−q)/2,(∆+p−p+q)/2) : We will make an ansatz for G 4 0,Õ by inspecting the OPEs between the lowest component of the multiplets of O and Φ. The ansatz for the G 4 1,Õ and G 4 2,Õ functions will come from the OPEs of the Q-descendants of the lowest component of the multiplets and (4.40), where we only need to change the dimension of the current multiplet for the dimension of the long multiplet. Since (4.39a), (4.39b) and (4.39c) have the same structures as (4.69a), (4.69b) and (4.69c), respectively, the OPEs for those cases are the same as the ones given in (4.42), (4.43) and (4.43), and we only need to replace the J and its Q andQ descendants by m p+p and its Q andQ descendants.
So far, we have not attempted to solve the constraint (4.66). Unlike the eigenvalue equations, we cannot expect to solve the constraint by each individual superconformal block, rather, we expect this constraint to fix the sum over all possible constraints, which in turn will imply relation among the crossing equations [22]. We expect this constraint to play a similar role as the constraints coming from the shortening conditions of the current multiplet, see (3.24) and (3.37).

Conclusions
In this article we have initiated a systematic way of computing the superconformal blocks for general four dimensional N = 1 SCFTs using the super Casimir operator. After constructing the most general four-point function with total vanishing U(1) R charge, we were able to compute and solve the super Casimir equations for correlators with one chiral and one anti-chiral field. Our results are in agreement with previous results involving only the superconformal blocks coming from the lowest component of the multiplets. Surprisingly, we have found that solutions to the s-channel super Casimir equations of the OO † ΦΦ correlator do not allow certain solutions to the three-point functions, setting their three-point function coefficient to zero, see (4.62) and the discussion below.
It was already know that studying the superconformal blocks of certain extended fourdimensional SFCTs imposed analytical constraints to some parameters of the theory, for example the central charge and the flavor central charge [11,38,39], but as far as we are aware of, this is the first time where consistency of the (super)conformal blocks implies restrictions to the three-point function coefficients of the theory. As we have mentioned above, a study of the superblocks for the lowest components is not enough to study the consistency of the super Casimir equations.
There are several interesting generalization of the present work. For example, we can study correlators with non-trivial C i functions. This will allow us to study the J J J J four-point function (3.50). The three-point function of two currents have, among other multiplets, an O (ℓ,ℓ+2) ∆ and an O (ℓ+2,ℓ) ∆ long multiplets, whose superblocks are identical if one only studies the superprimaries [8]. A complete analysis of the four point function will distinguish the contributions coming from the descendants. Including the C i functions will also allow us to check if the vanishing of λ (4) 4 in the three-point function of long multiplets (4.57) is a general feature, or if it is particular to the O O † ΦΦ correlator. Finally, including the D i and E 1 functions will allow us to study a four-point function solely composed by long multiplets. Another interesting generalization is to include non-trivial spin representations in our four point function. This will be a first step to write the superconformal blocks of the R-current multiplet [12]. Including long multiplets with non-zero spin might also impose constraints to the spectrum of the theory when including chiral fields.
Furthermore, most of our techniques are readily generalized to both extended supersymmetries and general dimensions. For example, in the case of extended supersymmetry, the four point functions ΦΦO 1 O 2 will depend only on B i type of functions for any four dimensional SCFT. This is due to the strong restriction imposed by the shortening conditions which will only be satisfied for the A 1 and B i functions. This statement will also be true when studying the four-point function for general dimensions, after taking care of the spinor indices for general dimensions, see [27,32] for a working example of this. We also point out that the stress-tensor multiplet in N = 2 theories, J N =2 , satisfies similar constrains as the current multiplet here studied [11,33,37], thus, we argue that the techniques presented in this article are easily generalized to include this multiplet in four-point functions. As has been shown in previous articles, four point functions including at least two stress-tensor multiplets yield strong bounds to the central charge and the flavor central charge [11,38]. Recently the superconformal blocks for the lowest component of the stress-tensor were found in [33]. We propose to use the super Casimir in order to obtain all the superblocks for this correlator and also for mixed systems including the moment map operators.
Finally, as we mentioned, it is possible to write G 0 0,O (4.9) as one superblock instead of a sum of several conformal blocks [40]. In order to do so, it necessary to study the asymptotic behavior of the superconformal partial waves using, for example, the supershadow formalism [9,40]. 13 To find such simple expression for the G 1,2,3,4 i,O superconformal blocks would greatly simplify computations, allowing for a more straightforward tool to solve the super Casimir equations.