$\mathcal{N}=1$ Superconformal Blocks for General Scalar Operators

We use supershadow methods to derive new expressions for superconformal blocks in 4d $\mathcal{N}=1$ superconformal field theories. We analyze the four-point function $\langle\mathcal{A}_1 \mathcal{A}_2^\dagger \mathcal{B}_1 \mathcal{B}_2^\dagger\rangle$, where $\mathcal{A}_i$ and $\mathcal{B}_i$ are scalar superconformal primary operators with arbitrary dimension and $R$-charge and the exchanged operator is neutral under $R$-symmetry. Previously studied superconformal blocks for chiral operators and conserved currents are special cases of our general results.


Introduction
Significant progress has been made recently in the conformal bootstrap program [1] for theories in higher than two spacetime dimensions . In particular, spectacular results have emerged from applying the bootstrap to supersymmetric systems [5,8,9,16,[22][23][24]29], where constraints from supersymmetry and knowledge of protected aspects of the spectrum make the approach even more powerful. A crucial ingredient in the superconformal bootstrap is the expansion of four-point functions in superconformal blocks, which sum up the contributions of all of the descendants of a given superconformal primary operator. Results for 4d superconformal blocks in N = 1, 2, 4 theories have previously appeared in [5,[30][31][32][33]24].
Recently, we introduced a new covariant approach to studying superconformal blocks [34], based on generalizing the shadow formalism developed in [35][36][37][38][39] to superconformal theories. In [34] we used this approach to analyze four-point functions of chiral and antichiral operators in theories with N = 1, 2 superconformal symmetry. In the present work we will apply our formalism to four-point functions containing general scalar operators in N = 1 theories, focusing on situations where the exchanged operator is neutral under the U(1) R symmetry.
The class of correlators we consider includes the interesting cases of chiral-antichiral four-point functions, for which the bootstrap was performed in [5,8,9], and also fourpoint functions of currents, which have been studied in [33,24]. These types of correlators (together with mixed chiral-current correlators which are also covered by our formalism) are extremely fruitful objects of study in the superconformal bootstrap for three reasons. Firstly, we have extensive knowledge of the protected spectrum of N = 1 superconformal theories. Secondly, four-point functions of scalars are currently the easiest systems for applying numerical bootstrap techniques. Thirdly, bootstrap techniques are often most powerful for four-point functions of low-dimension operators, and such operators are often protected. For these reasons, our expressions will likely be crucial ingredients in future explorations of the 4d N = 1 bootstrap.
The initial complication that arises in our analysis is the fact that multiple structures can appear in superspace three-point functions. Thus, our first task is to review the superembedding formalism for describing these structures and then to enumerate them, which we do in Sections 2 and 3. In Section 4 we set up and evaluate the superconformal integrals relevant for computing superconformal blocks, with our results given in Section 5. We also show how the cases of four-point functions containing chiral or conserved current operators emerge as special cases of our general result. In Section 6 we show explicitly how previous results for N = 2 superconformal blocks decompose into N = 1 superconformal blocks, providing a highly nontrivial consistency check on the form of the blocks. Several details of our calculations are presented in the appendices.
The basic superconformally covariant objects are supertwistors and dual supertwistors They transform as fundamentals and antifundamentals of SU(2, 2|1), so that the pairing Z Superspace is given by a pair of supertwistors Z a A , a = 1, 2, and a pair of dual supertwistors Z˙a A ,ȧ = 1, 2, subject to a constraint and with gauge redundancies Here, "∼" means "is equivalent to." This space has a natural action of the superconformal group given by matrix multiplication on the SU(2, 2|1) indices A. On the other hand, it is equivalent to the usual N = 1 superspace. To see why, one can choose the "Poincaré section" gauge fixing of GL(2, C) × GL(2, C), where (Z, Z) take the form The constraint (2.3) then reads x + − x − − 4iθθ = 0, so that we can identify x ± with the usual chiral/anti-chiral bosonic coordinates and θ, θ with the usual fermionic coordinates on superspace. Any function of Z's (Z's) alone is purely chiral (anti-chiral).
We will often work with bi-supertwistors which are invariant under the SL(2, C) × SL(2, C) subgroup of the gauge redundancies (2.4). A basic set of superconformal invariants are given by supertraces of products of X 's and X 's, for instance Here, p C denotes the fermion number parity of the index C (1 if C = 5, and 0 otherwise). 1 By construction, these invariants are chiral in unbarred coordinates and anti-chiral in barred coordinates.

Lifting N = 1 Fields to Superembedding Space
A four-dimensional N = 1 superconformal primary is labeled by its SL(2, C) Lorentz quantum numbers ( j 2 , j 2 ), its scaling dimension ∆, and its U(1) R charge R. It is convenient to summarize these labels as ( j 2 , j 2 , q, q), where the superconformal weights q, q are given by A general superfield with spin lifts to a multi-twistor operator in superembedding space (2.10) with homogeneity determined by its superconformal weights The field Φ is also subject to gauge redundancies in each index, and similarly for the other indices. It is convenient to introduce index-free notation by using auxiliary twistors S A , S A to absorb the indices of the superembedding fields. We define (2.13) The gauge-redundancy of Φ allows us to restrict S, S to be transverse and null 2 X S = 0, SX = 0, SS = 0. (2.14) Finally, the four-dimensional superfield is recovered by where the subscript "Poincaré" means we choose the Poincaré section gauge fixing (2.5).

Superconformal Integration
The superspace defined in Section 2.1 admits a natural notion of superconformally invariant integration. Note that the measure is superconformally invariant, and because of the delta function it is supported on Z, Z which satisfy the constraint (2.3). The form ω transforms in the following way under the gauge redundancies (2.4): ω → (det g)(det g)ω. (2.17) Suppose f (Z, Z) is a function that transforms oppositely under GL(2, C) × GL(2, C): Then the product ωf (Z, Z) is gauge-invariant and can be integrated, provided we divide by the volume of the gauge group This gauge-redundant integral is defined by the Faddeev-Popov procedure: we choose a gauge slice for GL(2, C) × GL(2, C), introduce the appropriate determinant, and integrate over the remaining variables.
Note that any f satisfying the condition (2.18) can always be written as a homogeneous function of X , X with degree −1 in both variables, so we will sometimes write f (X , X ) instead of f (Z, Z). We will also occasionally write D[X , X ] for D[Z, Z].
A special class of superconformal integrals will be particularly important in our computations. This is the case where f (X , X ) is independent of the fermionic variables so we may write f (X , X ) = f (X, X), where X σρ and X σρ are the restrictions of X AB and X AB to bosonic twistor indices σ, ρ = 1, 2, 3, 4. The 4 × 4 antisymmetric matrices X, X can also be thought of as six-dimensional vectors X, X ∈ C 6 transforming under SO(4, 2) ∼ = SU (2,2). 3 In [34], we showed that such superconformal integrals can be computed in a simple way in terms of non-supersymmetric conformal integrals, (2.21) Here, the notation ∂ 2 X means we take two derivatives with respect to X as an independent variable, and contract indices using the six-dimensional metric. 4 The conformally-invariant measure D 4 X on the right-hand side of Eq. (2.21) was defined in [39] and is given by 5 In this work, we will also encounter more general N = 1 superconformal integrals, where f (X , X ) is not independent of η, η and (2.21) doesn't immediately apply. For our purposes, we will be able to fix the required answers without going through a full computation. But for now, let us note how these integrals can be done in principle.
Consider a general function f (X , X ) and expand in the fermionic variables η, η. The measure ω contains a delta function where Z · Z is the SU(2, 2)-invariant pairing between the bosonic components of Z, Z. The presence of this delta function means that in any term with equal degree in η, η, we can replace leaving the integral unchanged. Meanwhile, terms with unequal degree in η, η integrate to zero. Thus, via the above replacement we can completely remove the η, η dependence of f (X , X ) and reduce to the case where (2.21) applies. 3 Our conventions are: In SU(2, 2) notation, this is ∂ 2 X ∝ ǫ αβγδ ∂ X αβ ∂ X γδ . 5 As written, the measures D[X , X ] and D 4 X are ambiguous under multiplication by an overall constant. This is because division by the infinite volumes vol(GL(2, C) × GL(2, C)) and vol(GL(1, C)) is really defined by a choice of Faddeev-Popov determinant in the gauge-fixing procedure, and the overall scale of this determinant is arbitrary. In this work, we choose determinants so that (2.21) holds. In other words, we absorb any difference in normalization of the two sides into the definition of D[X , X ]. The overall normalization of D 4 X will drop out of our computations and is unimportant (in practice we choose the same normalization as in [39]).

Supershadows
For an operator O(X , X , S, S) with quantum numbers ( j 2 , j 2 , q, q), we define the nonlocal shadow operator . The shadow transform (2.25) is uniquely determined up to a constant by the requirement that the integrand transform appropriately under GL(2, C) × GL(2, C) (2.18), together with the transverseness conditions on auxiliary twistors (2.14).
Within a correlation function, one can project onto the superconformal multiplet of O by inserting the projector In the definition above, | M schematically denotes a "monodromy projection" [39]. 6 Operationally, all we will need is the result for a monodromy-projected (non-supersymmetric) conformal integral, which has been derived previously [57,39] and is given in Eq. (4.16). The form of (2.26) is uniquely determined by superconformal invariance and the various conditions on O as a function of X , S, T and their conjugates.
The example of interest for us will be a four-point function The partial wave W O differs from the superconformal block G O by simple kinematic factors. In general the three-point functions A 1 A † 2 O and OB 1 B † 2 appearing in W O contain multiple structures, each with an independent coefficient. As we show explicitly below, W O will then receive independent contributions from each of these structures.

2-Point Functions
The two-point correlation function of a scalar superfield with its conjugate is determined by superconformal symmetry up to a constant. This fact is obvious in the superembedding 6 As detailed in [39], the monodromy projection restricts the region of integration in Eq. (2.26) so as not to interfere with the OPE expansion of the fields in a four-point function, thus avoiding extraneous "shadow" contributions in the computation of partial waves. space where there is only a single superconformal invariant with the correct homogeneity in X 1,2 , X 1,2 , where we have labeled the coordinates (X i , X i ) simply as (i, i). After restricting to the Poincare section, ij reduces to 1 2 Correlation functions of operators with spin are most easily expressed using index free notation. The two-point function between an N = 1 superfield and its conjugate is determined by superconformal symmetry up to a constant as This reproduces the superconformal two-point function when we project to 4d:

3-Point Functions
In this section we construct the superfield three-point correlation function is a real superfield with dimension ∆ and spin ℓ. Since superconformal invariance is explicit in superembedding space, a correlator is simply the most general function of independent invariants and tensor structures, consistent with the homogeneity properties of the participating operators. We will show that this correlation function contains 4 independent coefficients. In special cases such as A 1,2 being the chiral or conserved current superfields, there are additional constraints on the three-point function coefficients.
The general methods for constructing invariants and tensors in superembedding space are detailed in [41,34]. Here we show the relevant results. The three-point function This cross-ratio vanishes in the limit θ i = θ i = 0, i ∈ {1, 2, 0}, where ij = ji . It also has the following properties: In addition, the correlator can depend on two elementary tensor structures or equivalently where S12S denotes SX 1 X 2 S. The structure S + is nilpotent and satisfies the following relations: There are thus two independent spin-ℓ tensor structures:

General 3-Point Functions
The three-point correlation function A 1 A † 2 O in general contains four independent structures and can be written in the following form: The uniqueness of these structures follows from the relations (3.6, 3.9) above. 7 The λ However, if we impose shortening constraints on A 1,2 , such as making them chiral or conserved current multiplets, then there will be relations among the λ

Chiral Operators
In previous applications of the superembedding space and shadow formalisms, we've written down the chiral three-point function: This is not manifestly of the form (3.11). However, we can render (3.12) in the form we desire by using the identity 12 10 02 2δ = 12 21 10 02 01 20 (3.13) which follows readily from the definition (3.5), together with the fact that z is nilpotent of degree three. Specializing to δ = ∆+ℓ 4 , we obtain 01 10 02 20 ) 1 4 (∆+ℓ) . (3.14) Eq. (3.14) is now explicitly consistent with Eq. (3.11). The relative ratio between the four coefficients λ (i) ΦΦ † O is fixed by the chirality condition DȧΦ = 0. This alternative form will be useful when taking the chiral limit of our general superconformal block.

Conserved Currents
A conserved current superfield J has q J = q J = 1. This gives a three-point function .
(3.15) These coefficients are related to each other by the current conservation conditions D 2 J 1,2 = D 2 J 1,2 = 0, which impose the constraints However, the ratio between λ (0)

Shadow Operator and Correlation Functions
In order to calculate superconformal blocks we will also need the shadow three point function . This is given by applying Eq. (2.25): By superconformal symmetry, it must take the form 18) where T ± is defined analogously to S ± , and z = z 1→3,2→4 .
The shadow coefficients λ The relation between the coefficients may be determined by explicitly computing the integral in Eq. (3.17). The coefficients can also be uniquely fixed by requiring that the linear transformation respects the constraints for current and chiral 3-point functions and yields a partial wave consistent with unitarity. We give this argument explicitly in Appendix B. The resulting linear transformation is given by  Here we have normalized the shadow transformation so that applying it twice takes the three-point function coefficients back to themselves.

Partial Wave Computation
Now we will turn our attention to the four-point function, where as in the previous section the superfields A 1,2 and B 1,2 are Lorentz scalars with superconformal weights A 1,2 ∼ (q A , q A ) and B 1,2 ∼ (q B , q B ).
We wish to compute the superconformal partial wave W O corresponding to the exchange of a real, spin-ℓ superfield O ∼ ( ℓ 2 , ℓ 2 , ∆ 2 , ∆ 2 ) in the (12)(34)-channel. Up to overall normalization, this is given by inserting the projector (2.26) into the four-point function, and we have introduced the shorthand notation We will not attempt to evaluate the integral (4.2) in full generality. Rather, as in [34], we will focus on the case where the superfields in the four-point function (4.1) are restricted to their lowest component field. We refer to these superfields as the "external" fields, in constrast to the "exchanged" operator O. Setting the external fields to their lowest component means setting their Grassman coordinates, all to zero. This restriction is still of much interest, because the exchanged operator O remains a full-fledged superfield, and its associated partial wave is an essential ingredient for supersymmetric bootstrap applications. Note that setting θ ext to zero brings the integrand in (4.2) to a form where Eq. (2.21) applies.
The supertraces in Eq. (4.2) reduce as 8 In our index-free formalism, S + and T + represent matrices. Their proportionality to θ 0 θ 0 when θ ext = 0 can be seen, for instance, by going to a frame with x 1 → 0, x 2 → ∞ for S + and x 3 → 0, x 4 → ∞ for T + .
where the X i ∈ C 6 on the right are non-supersymmetric embedding vectors. With these observations in mind, the prescription in Eq. (2.21) then gives us (4.10) To compute the derivative in Eq. (4.9), it is natural to introduce the object This is the same quantity we encountered in our computation of superconformal blocks for the chiral four-point function in [34]. In Appendix A, we list several properties of N ℓ . For now, we simply note that when θ ext = 0, N ℓ (X 0 , X 0 ) → N ℓ (X 0 , X 0 ), and when 0 = 0 where C (λ) ℓ (x) are the Gegenbauer polynomials, a ℓ = 2 −6ℓ , and (4.13) In our notation, the (3 ↔ 4) acts on both the first term and the (1 ↔ 2) term.
It is natural to introduce N ℓ , because N (full) ℓ can be written in terms of N ℓ up to coordinate exchanges that take (1 ↔ 2) and/or (3 ↔ 4). In particular, recalling Eq. (3.10) for S ℓ − and S + S ℓ−1 − , we have (4.14) Similar formulas with the left-hand side multiplied by z's and z's are easily obtained by remembering that z is antisymmetric in (1 ↔ 2) and independent of 3 and 4, and vice versa for z.
At this point, an important simplifying observation is that W O | θext=0 is invariant under the simultaneous coordinate interchange 1 ↔ 2, 3 ↔ 4. Therefore, in Eq. (4.9), any piece of N (full) ℓ | θext=0 that is antisymmetric under this interchange must have vanishing contribution. By Eqs. (4.14), this is true for the following terms, and so we can ignore these terms from the outset. 9 This leaves us with where the dots denote terms that do not contribute to W O | θext=0 .
The remainder of the computation is straightforward. One inserts Eq. (4.15) into Eq. (4.9) and computes the ∂ 2 0 derivatives. This results in a sum over various conformal integrals, which are evaluated using the result of [57,39]: , (4.17) and g ∆ 12 ,∆ 34 ∆,ℓ (u, v) are the usual non-supersymmetric conformal blocks given by 10 Here, ∆ ij ≡ ∆ i − ∆ j , and g ∆ 12 ,∆ 34 ∆,ℓ is the conformal block for exchange of an operator with dimension ∆ and spin ℓ in a four point function of scalars with dimension ∆ i . The resulting expression for W O | θext=0 is a linear combination of the g ∆ 12 ,∆ 34 ∆,ℓ (as expected). The Additional details of the calculation are given in Appendix A.

Results
After relating the shadow coefficients to the coefficients of the original operator using Eqs. (3.19), the computations described in the previous section give the result where we have retained the overall OPE coefficient dependence to make it clear which structures contribute to each term. Here, g ∆,ℓ = g 0,0 ∆,ℓ is the conformal block for external scalars with ∆ 1 = ∆ 2 and ∆ 3 = ∆ 4 . The different terms present above reflect the decomposition of the superconformal multiplet of O into conformal multiplets [5].
If we take B 2,1 = A 1,2 , then λ and we obtain the superconformal block If there are no further constraints on λ (i) , then N = 1 superconformal symmetry cannot fix the relative coefficients between the supermultiplet of conformal blocks. However, additional symmetries or shortening conditions may impose interesting constraints on λ (i) . For the fourpoint function φφ † φφ † , where φ is the lowest component of a chiral multiplet Φ, we may plug in the three-point function coefficients in Eq. (3.14) to obtain: This agrees exactly with the previous results derived in [5] and provides a nontrivial check for our formalism.
Next let us consider the four-point function J 1 J 2 J 3 J 4 , where J i is the lowest component of a global symmetry current multiplet J i . This case was considered recently in [33,24]. The conservation condition D 2 J i = 0 imposes constraints on the three-point function coefficients as in Eqs. (3.16). Plugging in these relations we get: where the ratio is in general not fixed.
When J 1 , . . . , J 4 are identical conserved currents, then the four-point function is symmetric under permutations (x 1 ↔ x 2 ) or (x 3 ↔ x 4 ). This further constrains the three-point function coefficient. In particular, for even spin, λ For odd spin, λ J J O = 0 and we have: For the four-point function J 1 J 2 φφ † , we may plug in the conservation constraints for λ g ∆+1,ℓ−1 .

(5.7)
When J 1 and J 2 are identical currents, the even or odd spin blocks will pick up different parts of this result after setting λ The superconformal blocks in Eqs. (5.4-5.9) are in agreement with the expressions in the most recent version of [24] and the version of [33] to appear soon.
For the exchange of long multiplets O of dimension ∆, even spin ℓ, and vanishing R charge, Dolan and Osborn computed the contributions to be [30] The chiral block decomposition using these results was performed in [5], with the result Now we extend this result to the decomposition of G N =2|JJ;JJ and G N =2|JJ;φφ † . As explained in [24], the J ×J OPE can in general contain the descendants of unprotected N = 1 primaries with vanishing R-charge and (j, j) = ( ℓ±1 2 , ℓ∓1 2 ). In the present context, such operators arise as N = 2 descendants of the schematic form Q 2 Q 2 O, with dimension ∆ + 1. Because these operators have only one N = 1 descendant (which is a conformal primary) with vanishing R-charge and integer spin, up to a normalization factor the corresponding superconformal block is just a conformal block 2 ) ∝ g ∆+2,ℓ .
Similarly, for the decomposition of G N =2|JJ;φφ † we expect with no contribution from (6.4) because these operators cannot appear in φ × φ † . Matching to Eqs. (6.1) and (6.2) gives The existence of this decomposition provides a highly nontrivial check on the superconformal block results derived using our methods.

Summary and Outlook
We have computed the superconformal block G , where A 1,2 and B 1,2 are scalar superconformal primaries with general dimensions and R-charge, and the exchanged operator is R-charge neutral. When A 1,2 and B 1,2 are chiral, we reproduce the known result for superconformal blocks in the chiral-antichiral channel. Similarly, when A 1,2 are global symmetry currents and B 1,2 are either global symmetry currents or chiral operators, we obtain expressions for G There are many future directions to explore. Most immediately, the blocks we have computed provide new atomic ingredients to continue the study of N = 1 SCFTs using the numerical bootstrap. We also hope to apply supershadow methods to SCFTs with N > 1; for instance, the methods may be well-suited to study four-point functions of N = 2 supercurrents. Regarding the supershadow formalism in general, it would be interesting to develop further machinery that allows us to: (i) compute shadow integrals in a manifestly superconformally covariant way, rather than just a conformally covariant way as we have done in this work; and (ii) consider superconformal blocks where the exchanged primaries have nonzero R-charge. We hope to explore these various directions in the near future.

A Embedding-Space Derivatives and Integrals
In this appendix, we present some additional details of the calculation peformed in Section 4. We describe several properties of the quantities N ℓ , D ℓ , and z that are useful for computing the ∂ 2 0 derivative in Eq. (4.9). We present the results of this derivative acting on various terms, and evaluate the relevant conformal integrals.

A.1 N ℓ as a Gegenbauer Polynomial
One can show that the quantity N ℓ in Eq. (4.11) satisfies a recursion relation that identifies it as a Gegenbauer polynomial, where C Recall our notation, One can also write N ℓ as This expression follows from Eq. (A.1) after acting all the ∂ S,T derivatives.
When θ ext = 0, an SU(2, 2|1) trace reduces to an SU(2, 2) trace of the six-dimensional "sigma" matrices Γ m , Γ m . In particular, N ℓ (X 0 , X 0 ) reduces to a function of (X 0 , X 0 ). In this appendix, to be explicit, we define It is given by the same expressions as Eqs. (A.1) and (A.4), except that all auxiliary twistors and coordinates are reduced to their bosonic twistor parts. We write this simply as, for instance, S21034T → S21034T , etc.
With θ ext = 0, but prior to the ∂ 2 0 differentiation, 0 and 0 are considered to be independent. Afterwards, though, we identify 0 = 0, in which case Recalling that A.2 Symmetries of N ℓ , D ℓ , and z First, consider If we identify 0 with 0, then These expressions can be derived by using the Clifford algebra of the sigma matrices to commute 0 to the left or right and then using S0 = 0T = 0 (see Eq. (2.14)). It follows that S21034T is antisymmetric in 1 ↔ 2 and 3 ↔ 4, so N ℓ is either symmetric or antisymmetric in these exchanges depending on the parity of ℓ. This also follows from our expression for N ℓ as a Gegenbauer polynomial.
Similar formulas hold for z with 1, 2 replaced by 3, 4 respectively. In particular An important point is that z and z vanish when θ ext = 0 and 0 is identified with 0 (i.e. without supersymmetry, there is no conformally-invariant cross-ratio given three points). This simplifies derivatives involving these quantities, because the derivatives must act to eliminate all z's and z's.

B Shadow 3-Point Function Coefficients
In this appendix we will show how one can derive the transformation matrices in Eq. (3.19). Because 3-point function structures can only mix with structures with the same symmetry properties, the linear transformation λ Taking B 1 = B 2 = Φ to be chiral and using Eq. (3.14), we have the constraint     The above equations so far give 5 constraints on 8 unknowns.
The remaining freedom can be fixed by requiring that the superconformal partial waves are consistent with unitarity, i.e. that in reflection positive configurations the coefficients of individual conformal blocks have positive coefficients. Concretely, the computations described in Section 4 and Appendix A give the result In the reflection positive configuration B 1,2 = A 2,1 , each of the conformal block coefficients must be positive. This implies that where H depends on the overall normalization of the shadow transformation. A convenient choice is H = 1, which corresponds to defining the shadow transformation M(∆, ℓ) so that when applied twice it gives the identity M(∆, ℓ) · M(2 − ∆, ℓ) = 1 4×4 .