Bootstrapping Mixed Correlators in 4D $\mathcal{N}=1$ SCFTs

The numerical conformal bootstrap is used to study mixed correlators in $\mathcal{N}=1$ superconformal field theories (SCFTs) in $d=4$ spacetime dimensions. Systems of four-point functions involving scalar chiral and real operators are analyzed, including the case where the scalar real operator is the zero component of a global conserved current multiplet. New results on superconformal blocks as well as universal constraints on the space of 4D $\mathcal{N}=1$ SCFTs with chiral operators are presented. At the level of precision used, the conditions under which the putative"minimal"4D $\mathcal{N}=1$ SCFT may be isolated into a disconnected allowed region remain elusive. Nevertheless, new features of the bounds are found that provide further evidence for the presence of a special solution to crossing symmetry corresponding to the"minimal"4D $\mathcal{N}=1$ SCFT.


Introduction
The modern revival of the conformal bootstrap program [1] has led to remarkable progress in our understanding of conformal field theories (CFTs) in d > 2 spacetime dimensions. By studying the constraints of crossing symmetry and unitarity, it is possible to derive rigorous bounds on the scaling dimensions and operator product expansion (OPE) coefficients of any CFT. This approach relies on very few assumptions and can thus be used to study and discover theories without a known Lagrangian description.
A striking result of the numerical conformal bootstrap is that the bounds can develop kinks, or singularities, corresponding to known theories. This was observed in the 3D Ising [2] and O(N ) vector models [3] and was correlated with the decoupling of certain operators. This intuition was further developed in [4]. With the introduction of multiple correlators and additional assumptions on the number of relevant scalars, small regions surrounding the known theories can be isolated from other solutions of the bootstrap equations, i.e. the kinks become islands [5,6]. Consequently, the known theory is essentially the unique consistent solution of the crossing equations in a certain region in parameter space, given certain mild assumptions.
In d = 4 a kink was observed for N = 1 superconformal theories (SCFTs) with a chiral scalar operator φ [7][8][9]. More specifically, the scaling dimension bound for the first real scalar in thē φ × φ OPE develops a kink as a function of ∆ φ at the same point where the lower bound for the three-point function coefficient c φφφ 2 disappears. Similar behavior was also observed for theories in 2 ≤ d ≤ 4 with four supercharges [10]. In [9] it was conjectured that there is a 4D superconformal field theory (SCFT) that saturates the bootstrap bounds at the kink, referred to as the minimal 4D N = 1 SCFT. Based on the position of the kink and a corresponding local minimum in the lower bound on the central charge, this minimal theory was predicted to have c minimal = 1 9 and a chiral multiplet with scaling dimension ∆ φ = 10 7 , which also satisfies the chiral ring condition φ 2 = 0. Various speculations about this minimal theory have appeared [11]. In these proposals φ 2 = 0 is explicitly satisfied, but the central charge and the critical ∆ φ have not been successfully reproduced. As a result, the identity of this minimal theory remains elusive.
Motivated by this open problem, we study here the mixed correlator bootstrap for 4D N = 1 theories for the system of correlators { φ φφφ , φ RφR , RRRR }, where R is a generic real scalar and φ is a chiral scalar. We consider both the case where R is the first real scalar in theφ × φ OPE (beyond the identity operator of course), and that where R saturates the unitarity bound.
In the latter case it sits in a linear multiplet, which we will label by J. The bootstrap equations for the φ φφφ correlator were first considered in [7] and for JJJJ in [12], and for RRRR in [13]. Here we present new results for the superconformal blocks of φ RφR and φ JφJ . To be precise, we find superconformal blocks when the superconformal primary of the exchanged multiplet appears in a (j,) representation of SO (3,1), with j =. In this case the corresponding superconformal primary does not appear in the OPE of the external operators, but some of its superconformal descendants do. We also compute superconformal blocks of superconformal primaries in integer-spin representations; our results agree with the literature [13][14][15].
Our main results are new numerical constraints on 4D N = 1 theories. Studying the single correlator JJJJ , where J corresponds to a U(1) linear multiplet, we improve upper bounds on the OPE coefficients for JJJ and JJV where V is the spin-one multiplet containing the stress-energy tensor T µν . We also study these bounds as a function of the first unprotected scalar in the J × J OPE, deriving an upper bound on this operators scaling dimension and the JJO OPE coefficient. With the mixed correlator system for φ and R, with R the first real scalar in theφ × φ OPE, we will derive stronger lower bounds on the central charge c and upper and lower bounds on c φφR . In both cases we find interesting features near the minimal N = 1 point. Finally, studying the mixed correlator system for φ and J we will derive new bounds on c φφJ and cφ J(φJ) where (φJ) is the second scalar appearing in the φ × J OPE.
In sections 2 and 3 we give the complete set of conformal blocks for the mixed correlator system involving a generic real scalar multiplet R and the linear multiplet J respectively. In sections 4 and 5 we give the corresponding crossing relations for R and J. In section 6 we present results for the φ and R system. In section 7 we present results for the φ and J system. In appendix A we will go over the approximations used in the numerical implementation of the crossing equations and in appendix B we give some details on the derivation of the superconformal blocks.

Four-point functions, conformal and superconformal blocks
In this section we present our results for the superconformal block decomposition of the various four-point functions used in our bootstrap analysis. In particular we include results for the four-point function φ (x 1 )φ(x 2 )φ(x 3 )φ(x 4 ) , first obtained in [7,16], and new results for the four-point function φ (x 1 )R(x 2 )φ(x 3 )R(x 4 ) , with R a real operator, in theφ × R channel. In our numerical analysis we also use the four-point function φ (x 1 )R(x 2 )φ(x 3 )R(x 4 ) in theφ × φ channel, results for which were first obtained in [13] (see also [15]). This forces us to also consider , where again we use results of [13].
Four-point functions can be reduced and computed via the OPE. Consider the four-point where all operators are conformal primary. We can use the and similarly for ∆ kl , ∆ m , m is the scaling dimension and spin of the exchanged operator, and are the two independent conformally-invariant cross ratios constructed out of four points in space.
The conformal blocks g are functions that account for the sum over conformal descendants.
They are given by [17] 1 In N = 1 superconformal theories some of the conformal primaries in the sum in (2.1) are superconformal descendants, and so their contributions to the four-point function can also be accounted for by computing "superconformal blocks". The dimensions of the exchanged operators are constrained by unitarity to be [18] where ( 1 2 j, 1 2 ) is the representation of O under the Lorentz group, viewed here as SU(2) × SU(2), and q andq give the scaling dimension and R-charge of an operator via The four-point function φ ( involving the chiral operator φ and its complex conjugate can be expressed in terms of 12 → 34 contributions as [7] φ ( where we used cφ φO = (−1) c * φφO and with c 1 = ∆ + 4(∆ + + 1) , .
The unitarity bound here is ∆ ≥ + 2 and, when it is saturated, c 2 becomes zero.
where we used c φφO = c * φφO and (2.10) The difference between (2.7) and (2.10) is just in the sign of the g ∆+1, ±1 contributions.
In this work we will also decompose φ ( where we used c φφO = c * φφŌ and Gφφ ; φφ (2.12) In this case no superconformal block needs to be computed, but we need to include all classes of conformal primaries that can appear in the φ × φ OPE. This has been done in [16] and uses the fact that the product φ × φ is chiral and that the three-point function Φ Here z = (x, θ,θ) is a point in superspace, and the index I denotes Lorentz indices. The contributions we need to include turn out to be the superconformal primary φ 2 , protected even-spin operators of the formQO with dimension ∆ = 2∆ φ + , and unprotected even-spin operators of the formQ 2 O with dimension satisfying ∆ ≥ |2∆ φ − 3| + 3 + . When ∆ φ < 3 2 there is a gap in the dimensions of the unprotected and protected operators.

Four-point function
The four-point function φ (x 1 )R(x 2 )φ(x 3 )R(x 4 ) , involving the chiral operator φ and the real operator R, can be expanded in the 12 → 34 channel as where ∆ φ , ∆ R are the scaling dimensions of φ, R respectively, ∆, are the scaling dimension and , and we usē c φRŌ =c * φRO . As we will see below the sum in the right-hand side of (2.13) contains contributions from multiple classes of operators.
In order to compute Gφ R ; φR ∆, , ∆ φ −∆ R we need the general form of the three-point function Φ (z 1 )R(z 2 )O I (z 3 ) , where O I is a superconformal primary operator. To obtain this we use the results of [19,20]. To start, we note thatΦ has superconformal weights qΦ = 0 andqΦ = ∆ φ , while R has q R =q R = 1 2 ∆ R . General superconformal constraints imply that the three-point function is proportional to a function of X 3 , Θ 3 andΘ 3 [20], with the homogeneity property t I (λλX, λΘ,λΘ) = λ 2aλ2ā t I (X, Θ,Θ) , Quantities appearing in (2.14) are defined as withθ ij =θ i −θ j and the supersymmetric interval between x i and x j defined by Let us first assume that O I has q = 1 2 (∆ + ∆ φ ) andq = 1 2 (∆ − ∆ φ ), as would be the case if the zero component ofŌ I appeared in theφ × R OPE. Then, a =ā, which implies that t I in (2.14) can only be a function of the product Θ 3Θ3 . Furthermore, the Ward identity following from the antichirality property ofΦ implies that t I cannot be a function ofΘ 3 . Therefore, t I can only be a function of X 3 in this case.
With the constraints we just described the operator O I in (2.14) is an integer-spin tracelesssymmetric superconformal primary O α 1 ...α ;α 1 ...α , with the dotted and undotted indices symmetrized independently of each other, for which we can write where the dotted indices are symmetrized independently of the undotted ones. With (2.18) the θ expansion of both sides of (2.14) can be performed with Mathematica by extending the code developed for the purposes of [21]. We need the superconformal primary zero-components ofΦ and R, but then the possible contributions to the three-point function come not only from the zero component of O α 1 ...α ;α 1 ...α , but also from the conformal primaries in its θθ and θ 2θ2 components.
Taking into account all these contributions and using results of [21] leads to the superconformal blockḠφ R ; φR (2.20) The unitarity bound on O that follows from (2.4) is When the unitarity bound (2.21) is saturated, we see from (2.20) thatc 2 = 0 as expected. 2 The blockḠφ R ; φR ∆, , ∆ φ −∆ R we just computed constitutes merely one of the possible contributions to the right-hand side of (2.13). Further, we note that, in general, R is an operator exchanged in theφ × φ OPE, and so we also need to consider the three-point function Since Φ hasq = = 0, the unitarity bound (2.4) is modified to q ≥ j + 1. This implies that Φ has ∆ ≥ 1. In this case we only need to consider a conformal block g . Note that due to this contribution there is always a gap in the scalar spectrum of theφ × R OPE.
We should also consider the case where the zero component ofŌ does not contribute to thē φ × R OPE. Due to the antichirality property ofΦ it is still true that there cannot be aΘ 3 in t I , but now both Θ 3 and Θ 2 3 are allowed. In the first case, relevant operators are of the form O α 1 ...α ;αα 1 ...α for some and with , so that Q αŌ αα 1 ...α ;α 1 ...α is a spin-conformal primary that can appear in theφ × R OPE. 3 In this case and a superconformal block computation giveŝ (2.26) 2 As an aside we note here that, for a general scalar operator S with superconformal weights qS andqS , we get an expression similar to (2.19) for the corresponding blockḠφ S; φS ∆, , ∆ φ −∆ S , with the coefficients The blockĜφ R ; φR ∆, , ∆ φ −∆ R is another contribution to (2.13). We should note here that if the shortening condition Q (βŌαα 1 ...α );α 1 ...α = 0 is satisfied, then O is forced to haveq = − 1 2 ( + 1) [20]. As a result, the dimension of such O is fixed to be ∆ = ∆ φ − − 5 2 . This is below the unitarity bound ∆ ≥ ∆ φ + + 3 2 for this class of operators, but it nevertheless provides a check onĉ 2 of (2.26). 4 There is another case to consider with a Θ 3 , i.e. when we have a superconformal primary of Then, the conformal primary Q (α 1Ō α 2 ...α );α 1 ...α has spin and can contribute to theφ × R OPE. Corresponding to (2.14) we here have and the associated superconformal block iš (2.30) For operators O of this class such that Q αŌ αα 3 ...α ;α 1 ...α = 0, it follows that O hasq = 1 2 ( +1) [20]. This implies that the dimension of such O is ∆ = ∆ φ + − 1 2 , providing a check onč 2 of (2.30). 5 Note that this dimension of O is consistent with the unitarity bound for this class of operators needs to be considered. 4 For a general scalar operator S we get a blockĜφ S; φS ∆, , ∆ φ −∆ S similar to (2.25) but witĥ (2.27) 5 For a general scalar operator S we get a blockǦφ S; φS ∆, , ∆ φ −∆ S similar to (2.29) but witȟ The associated conformal block we have to include is g . The unitarity bound here is To summarize we may write, in (2.13), with the appropriate unitarity bounds, and with the contribution associated to (2.23) implicitly included in the first sum on the right-hand side.
Let us finally consider φ ( For the former we have where one contribution comes from As before, there are also contributions corresponding to superconformal descendants whose primary does not appear in theφ × R OPE. In particular, corresponding to (2.25) and (2.29) we havê while we also have theg conformal block contribution. The unitarity bounds are as explained above.
In the 14 → 32 channel we can use results of [13] to obtain where and In the 12 → 34 channel we can write Here the sum runs over superconformal primaries, but also over just conformal primaries if a superconformal primary does not contribute but one of its descendants does. Only evenspin operators can be exchanged in the R × R OPE. These can come from even-or odd-spin superconformal primaries, so that the sum in (2.40) runs over O 's with both even and odd spin.
, then, receives separate contributions from even-and odd-spin superconformal primaries. There are no constraints on R, except that it is a real operator of dimension ∆ ≥ + 2 by unitarity, and so from results of [13] we see that we cannot fix the coefficients of the conformal block contributions to the superconformal blocks. The best we can do is write and A superconformal primary that is not an integer-spin Lorentz representation can have superconformal descendant conformal primary components that contribute to (2.40). It turns out that we only need to consider superconformal primaries of the form O αα 1 ...α ;α 2 ...α with even ≥ 2 and q =q = 1 2 ∆. 6 The relevant operator is then the conformal primary contained in the superconformal descendantQ (α 1 Q α O αα 1 ...α ;α 2 ...α ) , where the undotted indices are the only ones that are symmetrized withα 1 . The conformal block we need to include is g ∆+1, with even ≥ 2 and ∆ ≥ + 3 by unitarity.

Four-point functions with linear multiplets
So far we have analyzed four-point functions including a chiral operator φ, its conjugateφ, and a real field R. The results we have obtained can be easily adapted to the case where the corresponding real superfield R is a linear multiplet J , containing a U(1) vector current j µ . Linear multiplets have q J =q J = 1, and appear in theories with global symmetries. The superspace threepoint function J (z 1 )J (z 2 )O(z 3 ) was considered in [22], where the superconformal blocks for were obtained in [12]. Our aim here is to obtain bounds using the system of correlators The associated superconformal-block decomposition of these four-point functions can be obtained from the results of section 2, given that J is a particular case of a real superfield with q J =q J = 1.
Since Q 2 (J) =Q 2 (J) = 0 and Q α (φ) = 0, we also need to make sure that the operators in the  (2.29), and (2.36), respectively, as well as g Using this we can define, in the 14 → 32 channel, where and Finally, in the 12 → 34 channel we can write and We should also mention here that there are conformal primary superconformal descendant operators that contribute to the four-point functions involving J, but whose corresponding superconformal primaries do not. This type of operators has been analyzed in detail in [12]. The result is that in order to account for these operators we need to include g ∆+1, with even ≥ 2 and ∆ ≥ + 3 by unitarity.

Crossing relations
Using the results of section 2 we can now write down the crossing equations that we use in our numerical analysis. It is well-known that from φ ( we obtain three crossing relations [8]. We get another three from φ ( , and a final crossing relation from R( . In total we have seven crossing relations.

RRO and
while, if is odd, c RRO = c
The crossing relation arising from φ ( and similarly forF,F.

System of crossing relations
The crossing relations (4.1), (4.3), (4.4), (4.9) and (4.11) can now be written in the form where the seven-vector V ∆, , ∆ φ , ∆ R contains the 3 × 3 matrices 14) and the remaining vectors are given by with definitions for X and X similar to that for X but involvingF,Ĥ,F,Ȟ, and + 1 independent nonvanishing derivatives, α i ∝ m,n a i mn ∂ m z ∂ n z 1/2,1/2 with m + n ≤ Λ. For example, for Λ = 17, a common choice in the plots below, the search space is 315-dimensional.

Crossing relations with linear multiplets
The crossing relations obtained in this case can be brought to the form where X ∆, 0, ∆ φ goes over just two scalar operators with dimension ∆ φ and ∆ φ + 2. Due to the determined coefficients in the superconformal blocks (3.4), (3.5), (3.7), and (3.8), the seven-vector V ∆, , ∆ φ contains 2 × 2 matrices now, contrary to the case in (4.13) where V ∆, , ∆ φ , ∆ R contained 3 × 3 matrices. Here, V ∆, , ∆ φ contains the matrices and the remaining vectors are given by with a similar definition for X , and The various functions F, F and H, H here are defined similarly to the analogous functions defined in section 4, using the superconformal blocks of section 3. We note that contrary to the case in section 4, the contributions of Z ∆, are not identical to those in V 7 ∆, , and so Z ∆, needs to be included in our numerical analysis.

Using only the chiral-chiral and chiral-antichiral crossing relations
A bound on the dimension of the first unprotected scalar operator R in theφ × φ OPE using just (4.1) was first obtained in [8] and recently reproduced in [9]. This bound, for Λ = 21 and Λ = 29, is shown in Fig. 1, and displays a mild kink at ∆ φ ≈ 1.4. The bound for Λ = 21 was first obtained in [8]. Here we provide a slightly stronger bound at Λ = 29.  Fig. 1: Upper bound on the dimension of the operator R as a function of ∆ φ using only (4.1). The generalized free theory dashed line ∆ R = 2∆ φ is also shown. The shaded area is excluded. In this plot we use Λ = 21 for the thin and Λ = 29 for the thick line.
If we assume that φ 2 = 0, then the allowed region on the left of the kink disappears [9,10], turning the kink into a sharp corner. The precision analysis of [9] suggests that the kink is at ∆ φ = 10 7 , although this relies on extrapolation. Using (4.1) we can also obtain a lower bound on the central charge. This is shown in Fig. 2 for Λ = 25. The corresponding bound for Λ = 21 first appeared in [8], and was later improved in [9].
The bound contains a feature slightly to the right of the kink of Fig. 1. Close to the origin the bound sharply falls just below the free chiral multiplet value of c = 1 24 in our normalization [7]. We may further assume that ∆ R lies on the bound of Fig. 1, and that R is the first scalar after the identity operator in theφ × φ OPE. The lower bound on the central charge obtained in this case is shown in Fig. 3. As we see, these extra assumptions strengthen the bound globally, but have the weakest effect around the free theory and ∆ φ ≈ 1.4. At that ∆ φ , which coincides with the position of the kink, we observe a local minimum of the lower bound on c. This result has also been discussed in [10], and is similar to the corresponding bound obtained in d = 3 in [2], although the free theory of a single chiral operator in our case has a lower c than the minimum in Fig. 3. The assumption φ 2 = 0 excludes the region to the left of ∆ φ ≈ 1.4. Therefore, we may conjecture that the putative theory that lives on the kink minimizes c among N = 1 superconformal theories that have a chiral operator φ that satisfies φ 2 = 0. Such theories were obtained recently [11] from deformations of N = 2 Argyres-Douglas theories [24], but they appear to have larger c than the one obtained for the minimal theory in [9], namely c minimal = 1 9 after extrapolating to Λ → ∞.

Using the full set of crossing relations involving φ and R
We will now explore bootstrap constraints using the full system of crossing relations (4.13). The virtue of considering mixed correlators is that they allow us to probe a larger part of the operator spectrum, e.g. we can obtain bounds on operator dimensions and OPE coefficients of operators in theφ × R OPE. In this subsection we assume that ∆ R lies on the (stronger) bound of Fig. 1. We also impose cφ Rφ = cφ φR -the implementation of this follows [6], i.e. we add a single constraint First we would like to obtain a bound on the OPE coefficient of the operatorφ in theφ × R OPE. We can obtain both an upper and a lower bound; they are both shown in Fig. 4. As we see  Fig. 4: Upper and lower bounds on the OPE coefficient of the operatorφ in theφ × R OPE as a function of ∆ φ , assuming ∆ R lies on the bound of Fig. 1 and demanding cφ Rφ = cφ φR . We also impose a gap equal to one between ∆ R and ∆ R . The shaded area is excluded. In this plot we there is a minimum of the upper bound slightly to the right of ∆ φ ≈ 1.4. Note that the bound of cφ Rφ at the minimum is lower than the free theory value which is equal to one.
Using mixed correlators we can also obtain a bound on the central charge similar to that of Fig. 3, i.e. assuming that ∆ R saturates its bound. The bound is shown in Fig. 5. As we see, even though we use the mixed correlator crossing relations the bound obtained is very similar to the corresponding bound in Fig. 3. The bound of Fig. 5 is weaker than that of Fig. 3 Fig. 5: Lower bound on the central charge as a function of ∆ φ , assuming that ∆ R lies on the bound of Fig. 1 and demanding cφ Rφ = cφ φR . We also impose a gap equal to one between ∆ R and ∆ R . The shaded area is excluded. In this plot we use Λ = 17.
With the inclusion of the crossing relations (4.3), (4.4) and (4.9) we can attempt to constrain scaling dimensions of operators with R-charge equal to that ofφ. In particular, we can attempt to find a bound on the dimension of the first scalar superconformal primary afterφ in theφ × R OPE, calledφ , assuming that ∆ R lies on the (stronger) bound of Fig. 1.
Numerically, this turned out to be a hard problem. For Λ = 11 a bound on ∆ φ did not arise for any value of ∆ φ . With the assumption that there are no Q-exact scalar operators in theφ × R OPE, i.e. neglecting the X and Y scalar contributions in (4.13), we managed to obtain a bound on ∆ φ but only for ∆ φ 1.12, after which point the bound was abruptly lost. This bound is shown in Fig. 6 Fig. 6: Upper bound on ∆ φ as a function of ∆ φ , assuming that ∆ R lies on the bound of Fig. 1 and imposing cφ Rφ = cφ φR . Here we neglect X and Y scalar contributions in (4.13), and impose a gap equal to one between ∆ R and ∆ R . The shaded area is excluded. In this plot we use Λ = 11.
Increasing our functional search space by taking Λ = 13, Λ = 17 and Λ = 19 we find a bound on ∆ φ up to ∆ φ ≈ 1.27, ∆ φ ≈ 1.32 and ∆ φ ≈ 1.34, respectively. At the corresponding ∆ φ the bound is again abruptly lost. Note that for these results we do not actually obtain the bound, but rather we ask if the spectrum withφ as the only scalar in theφ × R OPE is allowed or not.
We believe that numerical analysis for higher Λ will yield bounds on ∆ φ for higher ∆ φ , but it is puzzling that in going from Λ = 17 to Λ = 19 we have a very small gain in the ∆ φ up to which a bound on ∆ φ can be obtained.
The various features we have seen in plots of this section indicate the existence of a CFT with a chiral operator of dimension ∆ φ ≈ 1.4, or ∆ φ = 10 7 based on the analysis of [9]. Unfortunately the mixed correlator analysis has not allowed us to isolate this putative CFT from the allowed region around it, particularly from the allowed region for higher ∆ φ . We remind the reader that the region for ∆ φ < 10 7 can be excluded by imposing that φ 2 = 0 as a primary [9,10]. The set of conditions that isolate this putative CFT from solutions to crossing symmetry with higher ∆ φ have not been found in this paper. We hope that future work will be able to identify these conditions, or uncover a physical reason for their absence.

Using the crossing relation from JJJJ
Bootstrap bounds arising from the four-point function J(x 1 )J(x 2 )J(x 3 )J(x 4 ) were obtained recently in [12]. In fact, [12] considered the more complicated nonabelian case. Here we will consider just the Abelian case, where J carries no adjoint index, and obtain some further bounds that have not appeared before.
Since the dimension of J is fixed by symmetry, no external operator dimension can be used as a free parameter. For the plots in this section we will instead use the dimension of the first unprotected operator O in the J × J OPE as the parameter in the horizontal axis. Note that there is an upper bound to how large that dimension can get, and so our plots will not extend past that bound. This bound is found here by looking at the value for which the square of the plotted OPE coefficient turns negative.  The bounds in Figs. 7 and 8 were obtained using Λ = 29. 7 We can also obtain bounds for other values of Λ. We do this here letting O saturate its unitarity bound, i.e. choosing ∆ O = 2. The plots are shown in Fig. 9. As Λ gets larger we see observe an approximately linear distribution of the bounds, which we then fit and extrapolate to the origin. The fits are given by

Using the full set of crossing relations involving φ and J
Similarly to subsection 6.2 we can here obtain constraints on operators that appear in theφ × J OPE. One such operator isφ itself, and we can obtain a bound on its OPE coefficient. This OPE coefficient is equal to that of J in theφ × φ OPE, and its meaning has been analyzed in [7], where it was denoted by τ IJ T I 11 T J 11 . The bound is shown in Fig. 11 Fig. 11: Upper bound on the OPE coefficient of the operatorφ in theφ × J OPE as a function of ∆ φ , demanding cφ Jφ = cφ φJ . In this plot we use Λ = 17.
One application of this bound is in SU(N c ) SQCD with N f flavors Q i andQĩ. Mesons in this theory have scaling dimension ∆ M = 3(1 − N c /N f ), which can be close to one at the lower end of the conformal window, N f ∼ 3 2 N c . This was considered first in [7], where the meson M 1 1 was taken as the chiral operator and the relation was obtained for the contributions of the flavor currents of the symmetry group SU(N f ) L ×SU(N f ) R of SQCD. This satisfies our bound in Fig. 11 comfortably. For example, for N c = 3 and N f = 5, in which case ∆ M = 1.2, we have τ IJ T I 11 T J 11 ≈ 0.3 with the bound constraining this to be lower than approximately one. Even with these numerical results we are far away from saturating the bound with SQCD, although we can hope that by pushing the numerics further we will get much closer in the near future.
We should also note here that very close to ∆ φ = 1 our bound appears to be converging to a value for cφ Jφ below one, thus excluding the free theory of a free chiral operator charged under a U(1). While we have not been able to obtain a bound very close to one, i.e. 10 −15 or so away from it, we believe that the bound abruptly jumps right above one as ∆ φ → 1 in order to allow the free theory solution. This behavior of the bound has also been seen in [8].
As we have already seen the second scalar in theφ × J OPE has dimension ∆ φ + 2. We will call itφJ. We can obtain a bound on its OPE coefficient, again imposing cφ Jφ = cφ φJ . The bound is seen in Fig. 12, and is strongest close to ∆ φ = 1 where it approaches the expected value of cφ J(φJ) = 1.

Discussion
This work is the first numerical bootstrap study of mixed correlator systems in SCFTs with four supercharges. In this paper we focused on 4D N = 1 SCFTs and used the crossing symmetry and positivity in the { φ φφφ , φ RφR , RRRR } system, where R is a generic real scalar and φ is a chiral scalar. We also studied the special case with R → J, where J is the superconformal primary in a linear multiplet that contains a conserved global symmetry current. In all these cases we computed all necessary superconformal blocks, obtaining some new results.
We found new rigorous bounds on 4D N = 1 SCFTs that are stronger than those previously obtained. The features of our results strongly suggest the existence of a minimal 4D N = 1 SCFT with a chiral operator of dimension ∆ φ ≈ 1.4. Nevertheless, further studies are needed in this system of crossing relations. In particular, we did not find an isolated island of viable solutions to the crossing equations similar to that obtained in [5,6]. We believe that in order to address this more definitively we need to overcome the current practical limits on the dimension of the functional search space we can use with the available computational resources. When that becomes possible, we expect certain dimension bounds to become much more constraining. However, this will likely require a new level of both algorithmic efficiency and computational power. We expect to return to this system when such resource becomes available.
semidefinite programming techniques we have to approximate derivatives on F and F as positive functions times polynomials [8]. Here we explain how we do this for expressions like (A.1), assuming first that F contains a single conformal block. To signify this we will use F instead of where β and γ can here be either ∆ 1 − ∆ 2 or ∆ 2 − ∆ 1 depending on the four-point function we are considering, δ = 1 2 (∆ 1 + ∆ 2 ), and The constants α, β, γ, δ have specific relations to ∆, , ∆ 1 , ∆ 2 when appearing in (A.2), but below we will keep them general. As we see the crossing relation (A.1) takes a convenient form in terms of the function u β, γ, δ α (z). For our bootstrap analysis we now need to compute derivatives of u β, γ, δ α with respect to z orz, and evaluate them at z =z = 1 2 . An easy way to do this is to use a power series expansion. Indeed, the function u β, γ, δ α (z) can be expanded as C n α, β, γ, δ as given in (A.5) is nonpolynomial and thus not appropriate for our analysis. Hence, we take an alternate route here, based on that suggested in [7]. Using the hypergeometric differential equation it is easy to verify that u β, γ, δ α satisfies the differential equation If we use (A.4), then taking n − 2 derivatives on (A.6) and evaluating at z = 1 2 we find the recursion relation This allows us to write C n α, β, γ, δ = P n (α, β, γ, δ)2 δ−1 k β, γ α ( 1 2 ) + Q n (α, β, γ, δ)2 δ−1 (k β, γ α ) ( 1 2 ) , (A. 8) where (k β, γ α ) is the z-derivative of k β, γ α and the polynomials P and Q can be determined from (A.7).
is a polynomial in ∆, , β, γ, δ. In the case of H instead of F we find an expression similar to (A.11) but instead of the overall factor of 1 + (−1) m+n in (A.13) we have the factor 1 − (−1) m+n .

Appendix B. On the derivation of superconformal blocks
In this appendix we briefly describe the method we used to compute the superconformal blocks of section 2. Despite significant developments on N = 1 superconformal blocks [7,[12][13][14][15]22], blocks that arise from superdescendants whose corresponding primaries do not contribute have not been treated systematically. An example has been worked out in [22], while, in the case of interest for this paper, namely regarding theφ × R OPE, an example is the superconformal primary Oα, which cannot appear because it does not have integer spin, but whose descendantsQαOα and (the primary component of)Q 2 Q α Oα may both appear and form a superconformal block.
As mentioned in section 2, there are two types of such operators for the four-point function we are interested in. The first has = j + 1, that is, it has one more dotted than undotted . There is a second class of operators O α 1 ...α ;α 2 ...α , ≥ 1, that has one more undotted index.
In this appendix we summarize the calculation of such superconformal blocks in four-dimensional N = 1 SCFTs. We focus on the contribution of an exchanged superconformal multiplet in thē φ × R channel of the four-point function φ RφR . In d ≥ 3 dimensions, a superconformal multiplet includes a finite number of conformal multiplets. Therefore, the superconformal block is a linear combination of conformal blocks with coefficients fixed by supersymmetry. For each conformal primary component O of the supermultiplet, this coefficient is given by cφ RO c φRŌ /cŌ O , where cφ RO and c φRŌ are the three-point function coefficients and cŌ O is the two-point function coefficient.
The construction of primary components and their two-point function coefficients cŌ O for any 4D N = 1 superconformal multiplet has been worked out in [21]. The form of the superfield three-point function was originally worked out in [19,20], and reproduced for the cases of interest here in (2.14), (2.24) and (2.28). Using the Mathematica package developed in [21], we expand these three-point functions in θ andθ. Using the explicit construction of the superfield at each θ,θ order worked out in [21], we match the result of the expansion of the superfield three-point functions to the expected form of conformal three-point functions and solve for the three-point function coefficients cφ RO .
For the second class of operators we carried out a similar procedure and obtained (2.29) and (2.30).
Although we will not present the details here, this calculation is easily generalized to cases where the operator R is not real and carries an R-charge.