Exploring the Minimal 4D $\mathcal{N}=1$ SCFT

We study the conformal bootstrap constraints for 4D $\mathcal{N}=1$ superconformal field theories containing a chiral operator $\phi$ and the chiral ring relation $\phi^2=0$. Hints for a minimal interacting SCFT in this class have appeared in previous numerical bootstrap studies. We perform a detailed study of the properties of this conjectured theory, establishing that the corresponding solution to the bootstrap constraints contains a $\text{U}(1)_R$ current multiplet and estimating the central charge and low-lying operator spectrum of this theory.

Hints for a possible new 4D CFT with N = 1 supersymmetry appeared in [16] (building on earlier studies [17][18][19][20]), manifesting as a kink in general bounds on the scaling dimension of the leading non-chiral scalar in the OPEφ × φ, where φ is a chiral operator. This coincided with the disappearance of a lower bound on the chiral operator OPE coefficient φ × φ ∼ λ φ 2 φ 2 , allowing this coefficient to vanish precisely at this dimension. Moreover, it was established in [9] that a similar feature appears at all 2 ≤ D ≤ 4 in SCFTs with four supercharges, where as D → 2 it merges with the 2D N = 2 minimal model. The absence of the φ 2 operator in D = 3, 4 could also be seen more directly in the approximate solutions to crossing symmetry reconstructed in [9].
However, the correct interpretation of these features in both D = 3, 4 is not yet understood.
Based on their similarity to features that are known to coincide with the 3D Ising [1][2][3] and 3D O(N ) vector models [4,6], it is tempting to conjecture the existence of a family of new SCFTs. In this work we study the 4D N = 1 version of these kinks in greater detail, exploring the properties of the theory that we conjecture to live there.
We will establish several properties of this conjectured theory using the conformal bootstrap conditions for the correlator φ φφφ , building on the earlier results of [9,16]. First we establish directly that assuming the chiral ring condition φ 2 = 0 imposes a sharp lower bound ∆ φ ≥ 1.415.
In particular we exclude the possibility that ∆ φ = √ 2. Second, after imposing the chiral ring condition we place a bound on the leading spin-1 superconformal primary and find that it forces the existence of a U(1) R current multiplet when the lower bound on ∆ φ is saturated.
Having established that this putative theory contains a U(1) R current multiplet (whose descendant is the stress-energy tensor), we proceed to compute general lower and upper bounds on the conformal central charge for SCFTs with φ 2 = 0. The upper bounds are somewhat dependent on the gap until the next spin-1 primary, but for all gaps the lower and upper bounds merge at the minimal value of ∆ φ . We estimate that this minimal theory has c/c free 8/3 where c free is the central charge of a free chiral multiplet. We also make preliminary determinations of the OPE coefficient of theφφ operator, the dimensions of the second scalar and spin-1 superconformal primaries, and the dimension of the leading spin-2 superconformal primary.
In the present work we have not yet found a set of gap assumptions that isolate this solution, i.e. we do not yet see islands analogous to what was found in [2,3,6]. For this we anticipate that we will need to consider a larger system of correlators containing both the φ andφφ operators.
However, in our current setup we can already uncover a lot of information about this theory and we hope that the results of this paper are a useful step towards identifying the nature of this mysterious 4D N = 1 SCFT.

Results
In this work we study the correlator φ φφφ where φ is a chiral operator in a 4D N = 1 SCFT, similar to what was done in [16]. Crossing symmetry of this correlator leads to the sum rules where the functions of the conformal cross-ratios z andz that appear are related to conformal and superconformal blocks and defined in [16]. 1 In general we assume that the superconformal primary operators O in the first sum satisfy the unitarity bound ∆ O ≥ + 2 [23], while the even-spin operators in the second sum may either be conformal primaries in BPS multiplets with ∆ O = 2∆ φ + , or conformal primaries in unprotected multiplets satisfying the unitarity bound 18,20].
As described in [16], in order to rule out assumptions about the spectrum we can look for a 3-vector of functionals α that when applied this sum rule leads to a contradiction. In particular, if the functional is > 0 on the identity operator contribution and ≥ 0 on all other possible contributions, then the sum rule can never be satisfied. Alternatively, by normalizing the functional on the contribution of a particular operator O 0 and extremizing the action on the identity operator, we can obtain upper or lower bounds on the OPE coefficient of O 0 . We apply this logic below to obtain bounds on operator dimensions and OPE coefficients, using SDPB [3] to solve the relevant optimization problem after phrasing it in the language of semidefinite programming. The functional search space is governed by the parameter Λ, where each First, we reproduce the general upper bound on the dimension of the leading unprotected scalar operator ∆φ φ , finding precise agreement with [16]. This bound is shown in Fig. 1 there is a mild kink in this upper bound around ∆ φ 1.407. We show how this position changes as we increase the search space of the functional in a later plot. Note that any theory saturating Next we recompute this bound imposing the additional condition that the chiral φ 2 operator does not appear in the φ × φ OPE. This condition has the effect of excluding all points to the left of the dotted vertical line in Fig. 1. The region to the right remains the same. In other words, it imposes the strict lower bound ∆ φ ≥ 1.407, causing the mild kink to turn into a sharp corner.
One can see that this had to be the case by considering bounds on the OPE coefficient of the operator φ 2 , shown in Fig. 2. The lower bound on λ φ 2 disappears exactly at ∆ φ = 1.407. Thus, Fig. 2 makes it clear that if we demand λ φ 2 = 0, implying that φ 2 is not in the spectrum, then all points to the left of ∆ φ = 1.407 must be excluded. Our general bound is also compatible with the results of [9], which found that the φ 2 operator was absent in approximate solutions to crossing symmetry living on the boundary of the allowed region to the right of the kink.
In Fig. 3 we show an upper bound on the OPE coefficient λφ φ of an operator whose dimension saturates the bound in Fig. 1. Without any additional assumptions the upper bound attains a minimum at precisely the location of the kink, occuring at λφ φ 0.905. If we further impose the absence of φ 2 , then all points to the left of the dotted vertical line in Fig. 3 are excluded.
Next we would like to ask the question: if there is an SCFT living near the kink with the chiral ring relation φ 2 = 0, does it contain a stress-energy tensor? In other words, could it correspond to a local SCFT? In Fig. 4 we assume φ 2 = 0 and place an upper bound on the leading spin-1 superconformal primary V in theφ × φ OPE, again at Λ = 21. We see that the bound on ∆ V approaches 3 as ∆ φ approaches its minimum value. Thus, the U(1) R current multiplet V R is required to be in the spectrum at this point.
Note that for sufficiently small ∆ φ the bound excludes the line that would correspond to a generalized free theory with ∆ V = 2∆ φ + 1. This is natural, as our assumption that φ 2 is absent is not true in a generalized free theory. On the other hand, when ∆ φ ≥ 3/2, the contribution in the sum rule corresponding to the chiral φ 2 operator is identical to one contained in the unprotected scalar contributions in φ × φ. Thus, we expect the generalized free line should be allowed for ∆ φ ≥ 3/2. Here we see that it crosses this line at ∆ φ ∼ 1.486, compatible with this expectation. Now that we have established the existence of a U(1) R current multiplet, we can assume it to be in the spectrum and place an upper bound on the second spin-1 operator V . The result is shown in Fig. 5. We see that ∆ V 4.25 at the minimum value of ∆ φ .
We can also compute general lower bounds on the central charge c, using that the OPE coefficient λ 2 V R = ∆ 2 φ /72c. Here our normalization is such that c free = 1/24 for a free chiral multiplet. Similar bounds were computed in [16]. Here these bounds are shown in Fig. 6 for Λ = 21, 23, . . . , 29. As in [16], these bounds drop very sharply as ∆ φ → 1 so as to be compatible with the free theory value c free = 1/24.
We can also impose a gap until the second spin-1 dimension ∆ V and find upper bounds on c for each value of the gap. These bounds are shown in Fig. 7 at Λ = 21, where we have also imposed that there is no φ 2 operator. We see that the upper and lower bounds meet at the minimum value of ∆ φ , essentially uniquely fixing the central charge at this point, with c .081 at Λ = 21.
On the other hand, as seen in Fig. 6 has not yet completely converged at Λ = 35, but there is a striking linear relation between ∆ φ and c, given approximately by c ≈ 1.454∆ φ − 1.965. Moreover, as we increase Λ the rate of convergence appears to be well-described by a fit that is linear in 1/Λ (similar to the fit done in [15]), These fits are shown in Fig. 9. While these extrapolations should be taken with a grain of salt, it is intriguing that the minimal point may be converging to c(∞) = 1/9 or c(∞)/c free = 8/3. If the minimal 4D N = 1 SCFT exists and has a simple rational central charge, this is our current best   conjecture. 2 It is also possible that ∆ φ is converging to the rational value ∆ φ (∞) = 10/7.
We finish with some preliminary explorations of the higher spectrum. In Fig. 10  upper bound on the dimension of the second nonchiral scalar inφ × φ, assuming that the first saturates its upper bound and also assuming the chiral ring relation φ 2 = 0. Based on this we obtain the estimate ∆ R 7.2.
In Fig. 11 we show an upper bound on the leading spin-2 superconformal primary inφ × φ assuming φ 2 = 0 in the chiral ring. At least at Λ = 21, this bound is very close to the generalized free value when ∆ φ attains its minimal value, ∆ S 4.82. We do not know why this is the case, given that the chiral ring relation does not hold in the generalized free solution and we could potentially exclude this line for ∆ φ < 3/2.
It will be interesting in future studies to see how much of these allowed regions are compatible with the conditions of crossing symmetry for larger systems of correlators-in particular we would  Fig. 11: Upper bound on the dimension of the leading superconformal primary spin-2 operator in the OPEφ × φ as a function of the dimension of φ. The shaded area is excluded. Everything to the left of the vertical dotted line at ∆ φ = 1.407 is excluded due to the assumption that there is no φ 2 operator. The generalized free theory dashed line ∆ S = 2∆ φ + 2 is also shown. In this plot we use Λ = 21.
like to know whether our minimal solution survives and can be isolated e.g. using the condition that the φ ×φφ OPE contains a gap between φ and the next scalar operator. We hope that pursuing a mixed correlator study will lead to small islands similar to what was found in [2,3,6].
It would also be interesting to see if there are corresponding minimal theories with more general chiral ring relations φ n = 0. We hope to pursue these directions in a future study.
If this solution survives, the crucial question is to identify the underlying nature of this theory.
The small central charge c 1/9 indicates that this theory must have a very small amount of matter and this is not very easy to accomodate in asymptotically-free 4D gauge theories. For example, N = 1 SQCD theories all have central charge larger than 1. The properties of this theory are similar to Wess-Zumino models with a W = φ 3 superpotential, but it has been known for a long time that such theories do not have an interacting fixed point in 4D [25]. Thus, it may be that we have stumbled across a new non-Lagrangian N = 1 SCFT. It would be interesting to better understand if it could arise as a deformation of a known non-Lagrangian theory such as one of the Argyres-Douglas fixed points [26], or perhaps by coupling a known N = 1 SCFT to a topological field theory [27]. We leave further exploration of these possibilities to future work. thank the Aspen Center for Physics for hospitality during the completion of this work, supported by NSF Grant No. 1066293. The computations in this paper were run on the Omega and Grace computing clusters supported by the facilities and staff of the Yale University Faculty of Arts and Sciences High Performance Computing Center.