Landscape of Simple Superconformal Field Theories in 4d

We explore the space of renormalization group flows that originate from $\mathcal{N}=1$ supersymmetric $SU(2)$ gauge theory with one adjoint and a pair of fundamental chiral multiplets. By considering all possible relevant deformations - including coupling to gauge-singlet chiral multiplets - we find 34 fixed points in this simple setup. We observe that theories in this class exhibit many novel phenomena: emergent symmetry, decoupling of operators, and narrow distribution of central charges $a/c$. This set of theories includes two of the $\mathcal{N}=2$ minimal Argyres-Douglas theories and their mass deformed versions. In addition, we find 36 candidate fixed point theories possessing unphysical fermionic operators -including one with central charges $(a, c)\simeq (0.20, 0.22)$ that are smaller than any known superconformal theory -that need further investigation.


INTRODUCTION
Conformal field theory (CFT) is an important object in theoretical physics, which displays the physics of the low energy fixed points of some gauge theories and of critical phenomena in condensed matter theories.One interesting question of CFT is to find the "minimal" interacting theory.In four dimensions, a measure of minimality is the a central charge, the coefficient to the Euler density of the trace anomaly, due to the a-theorem [1,2], a UV > a IR for all unitary renormalization group (RG) flows.A related quantity is the c central charge, the coefficient to the two-point function of the stress-energy tensor.
However, it is difficult to analyze strongly-coupled CFTs, even a and c, in general.The conformal bootstrap program [3] partially solves this by giving a bound on c, but does not tell what the actual minimal theory is.The situation changes drastically in theories with supersymmetry.The superconformal symmetry allows to relate the central charges to 't Hooft anomalies of the R-symmetry [4], which are determined by the amaximization [5] or higher supersymmetry itself.
The central charge c of any unitary interacting N = 2 SCFT satisfies c ≥ 11  30 [6].The theory that saturates the bound is the Argyres-Douglas theory [7,8], denoted as H 0 or (A 1 , A 2 ) in the literature.H 0 also has the smallest known value of a for an interacting N = 2 theory.
In N = 1 theories, no analytic bound on the central charges is known so far.However, the numerical bootstrap suggests that the SCFT with the minimal central charge has a chiral operator O with chiral ring relation O 2 = 0 [9][10][11], and a bound c ≥ 1/9 0.11 [11].Is there a theory which saturates this bound?The minimal theory thus far known in the literature has a = 263 768 0.34 and c = 271 768 0.35, which was constructed via a deformation of the H 0 theory [12,13], and thus denoted as [14] This value of c is large compared to the bound.
In the present work, we initiate a classification of N = 1 SCFT in four dimensions obtained from Lagrangian theories to find a minimal SCFT.We explore the space of RG flows and fixed points that originate from the simple starting point of supersymmetric SU (2) gauge theory with one adjoint and a pair of fundamental chiral multiplets.From this minimal matter content, we consider all the possible relevant deformations, including deformations by coupling gauge-singlet chiral multiplets.Among the fixed points we obtain, two have enhanced N = 2 supersymmetry: the Argyres-Douglas theories H 0 and H 1 , as already found in [15][16][17].The other 32 are N = 1 supersymmetric, including the H * 0 theory as a minimal theory in terms of a.We verify that these are "good" theories in the sense that there is no unitaryviolating operator by utilizing the superconformal indices [18,19].It is remarkable that (a, c) of these SCFTs distribute within a narrow range as in Figure 1, although the allowed bound of a/c is wide [20].
In addition, we find 36 candidate fixed points which have an accidental global symmetry in the infrared and some unphysical operators, thus we refer to them as "bad" theories.Remarkably, these include theories with even smaller central charges than those of H * 0 .The minimal one, which we denote as T M , has a 0.20, and c 0.22.Although we are not able to conclude that these bad theories are really physical by the present techniques, we scope their properties.
2. Find the set of all the relevant chiral operators of T , which we will call R T .Let us also denote S T ⊂ R T as the set of operators with R-charge less than 4/3. 5.For each of the new fixed point theories obtained in previous steps, check if it has an operator O d that decouples.Remove it by introducing an N = 1 chiral field X and a superpotential coupling δW = XO d .We will use this notation to clearly distinguish X from M in the following.

Consider the fixed points {T
6.For each new fixed point, repeat the entire procedure.Terminate if there is no new fixed point.
We employ the a-maximization procedure [5] and its modification [21] to compute the superconformal Rcharges at each step.Beyond a-maximization, we check whether the theory passes basic tests as a viable unitary SCFT: one is the Hofman-Maldacena bounds for N = 1 SCFTs, 1 2 ≤ a c ≤ 3 2 [20]; the other one is the superconformal index.Some of the candidate fixed points have trivial index, or violate the unitarity constraints [22,23].
We perform this procedure for SU (2) gauge theory with the adjoint chiral multiplet φ and two fundamental chiral multiplets, q and q (N f = 1).When there is no superpotential, this theory flows to an interacting SCFT T , as discussed in [24] (also see [25]), and a free chiral multiplet Trφ 2 .To pick up only the interacting piece, we add the additional singlet X and the superpotential Starting from T , we apply the deformation procedure, and find 34 non-trivial distinct fixed points.These theories pass every test we have checked, so we call them "good" theories.One caveat is that some of these "good" theories have a flavor symmetry that is not classically manifest.This can be explicitly seen from the superconformal index as we will discuss shortly.In these cases we cannot rule out the possibility that this symmetry mixes with the superconformal R-symmetry causing unitarity violation.Nevertheless we keep referring to these theories as "good", assuming there is no such mixing.We list the theories with this feature in the appendix.
There are an additional 36 distinct theories that pass almost all of our checks, except that there is a term in the index that signals a violation of unitarity.The existence of such a term implies that either the theory does not flow to an SCFT in the IR, or the answers we obtained were incorrect because we failed to take into account an accidental symmetry.In fact, these "bad" theories also have an accidental U (1) symmetry which is not visible at the level of the superpotential, but is evident by the existence of the corresponding conserved current term present in the index.At present we do not know how to account for this accidental symmetry, and so cannot say for certain if these flows will lead to SCFTs or not.
Interestingly, 6 of these "bad" theories appear to have central charges lower than that of H * 0 .Denote the lowest one T M .This is a hint that there might be a minimal SCFT in this landscape.
We have plotted a, c for the "good" theories without this interesting complication in Figure 1.We see that the distribution of a vs c are concentrated near the line of a/c ∼ 0.87.All the theories satisfy the Hofman-Maldacena bound, and more curiously the stronger bound conjectured by [26] 3 5 ≤ a c .Of the "good" theories, H * 0 has the smallest value of a. T 0 has the smallest value of a among any theory with a U (1) flavor symmetry.H * 1 has the smallest value of a among any theory with an SU (2) flavor symmetry [27].Below we examine each of these "minimal" theories in turn, as well as the lowest central charge theory T M , and the second-to-lowest a central charge "good" theory with no flavor symmetry, which we denote T µ .We summarize the structure of RG flows among these special theories in Figure 2. The full list of our theories appears in the appendix.
The superconformal indices of these theories can be computed using the Lagrangian description.We define the index as where (j 1 , j 2 ) are the spins of the Lorentz group and r the U (1) R-charge.When the theory has a global symmetry with Cartan generator f , we also include the fugacity x for it.For each of these special theories, we give the first A subset of the fixed points that can be obtained from SU (2) N f = 1 adjoint SQCD with singlets.The arrows are labeled by the superpotential deformations.Note that the graph is not arranged vertically by decreasing a central charge, because the deformations we consider involve coupling in the singlet fields.
few terms in the reduced superconformal index which removes the conformal descendant contributions coming from spacetime derivatives.If the reduced index contains a term t R χ 2j+1 (y) with R < 2 + 2j or a term (−1) 2j+1 t R χ 2j+1 (y) with 2 + 2j ≤ R < 6 + 2j, it violates the unitarity constraint [22,23].The coefficient of t 6 y 0 allows us to read off the number of marginal operators minus the number of conserved currents [22].The superpotential F-terms ∂W/∂ϕ = 0 for the fields ϕ allow us to read off the classical chiral ring, and quantum modifications can be argued from the index.We will see that the chiral rings we study in this paper are subject to the quantum corrections.The superconformal index turns out to be a useful tool to study the fully quantum corrected chiral rings of our models.
Let us begin with the T 0 SCFT which is obtained via a deformation of T , The IR R-charges of the fields of the T 0 and all other theories discussed below are given in Table I.This theory has the second smallest value of a, and the smallest value of c among the 34 "good" fixed points we find [28].
The chiral ring of the theory can be easily studied: the F-term conditions from (3) are simply Trφ 2 = 0, qφ = 0 and Xφ + q 2 = 0.The first equation truncates the chiral ring by setting φ 2 = 0.The second and third equations lead to the classical generators of the chiral ring: O ≡ Trq q, Trφq q and X, with relation O 2 ∼ XTrφq q.
This theory has an anomaly free U (1) flavor symmetry that mixes with R. The reduced index is given as I r (t, y; x) = t 3.28 x 12 − t 3.45 x −2 χ 2 (y) + t 4.19 x 8 − t 6 + t 6.56 x 24 + t 7.46 x 20 + t 8.27 x −10 + . . ., where we assigned the flavor charges for the fugacity x as f q = 1, Here and below χ s (a) denotes the character for the SU (2) flavor symmetry of dimension s = 2j + 1.This index allows us to read off the quantum modified chiral ring: the terms t 3.28 x 12 and t 4.19 x 8 in the index come from the chiral operators Trφq q and Trq q respectively; the second term denotes the fermionic operator O α = TrφW α .We see that the operator X (which would contribute t 5.10 x 4 to the index if it exists) is absent from the chiral ring [29].We can read off the chiral ring relation O 2 = O α • (Trφq q) = 0 from the absence of the terms t 8.38 x 16 and −t 6.73 χ 2 (y)x 10 .
The H 0 fixed point can be obtained from T 0 by adding the M Trφq q term.This superpotential is indeed a simplified version of the one considered in [15].At the H 0 fixed point we further deform by a mass term M 2 , This flows to the H * 0 theory with the central charges The H * 0 SCFT has been studied in [12,13] as a deformation of the H 0 Argyres-Douglas theory.Utilizing the UV Lagrangian description presented here, we are able to confirm various predictions about H * 0 .Classically, the F-terms of (6) imply that M, X, and O ≡ Trq q generate the chiral ring, with relations M 2 ∼ 0 and O 2 ∼ 0. The superconformal index for the H * 0 theory can be computed to give a reduced index From this we see that the two generators M and O contribute the t 3 and t 4 respectively, while X is not a generator.We also find that the operator O α = Tr(φW α ) contributes to t 7 2 χ 2 (y).From the coefficients of t 6 , t 7 , t 8 , we find M 2 = M O = O 2 = 0 in the chiral ring.The term t 7 comes from (O α ) 2 .There is a relation for O α of the form M O α = O O α = 0 which can be read from the absence of the terms −t 13 2 χ 2 (y) and −t 15 2 χ 2 (y).These relations support the analysis of [12,13].
The flow to H 1 in our setup is a simplified version of the flow considered in [16], and was also considered in [30].From H 1 the H * 1 SCFT is then obtained via a mass deformation to the singlet, The central charges are We see the theory has the SU (2) current from the −t 6 χ 3 (a) term, which is visible at the level of the superpotential.There are generators M , X and O i satisfying the relations There are also fermionic operators O α = Tr(φW α ) with relations M O α = XO α = 0.

Tµ -NEXT TO MINIMAL
The T µ SCFT is obtained by the superpotential and the central charges are given by a Tµ = 711 2048 0.3472, c Tµ = 807 2048 0.3940.(13) The value of a is the third smallest value among the "good" theories we find and the second among the ones without flavor symmetry.The reduced index is We see the chiral ring relations (M 1 ) 3 = (M 2 ) 2 = (M 3 ) 2 = 0.There is also an operator O α = Tr(φW α ) with a relation M 2,3 O α = 0.There is no flavor symmetry as we do not have the −t 6 term nor a marginal operator.

TM -A NEW MINIMAL THEORY?
Let us discuss one example among the 30 "bad" candidate fixed points.Consider the superpotential There is no anomaly-free flavor symmetry in the Lagrangian.Assuming the R-charges are fixed by W and anomaly condition, we get the central charges as The term −t 6 implies that there is a conserved current at the fixed point if it exists.The term t 21 4 χ 2 (y) violates the unitarity bound.As far as we know, this phenomenon has not been discussed in the literature.One possibility is that this term comes from the short multiplet C− 1 4 (0, 1  2 ) (in the notation of [22,31,32]) that becomes free and gets decoupled along the RG flow.Subtracting the contribution to the central charges by treating the bottom component as a free fermion with R = −1/4, we get the central charges (a, c) = ( 189 1024 , 189 1024 ) (0.1846, 0.1846).We do not know if this prescription yields the correct central charges nor index of the IR theory.
Even though we do not have a valid index, if we take it literally we can read off the chiral ring relations M 2 = 0, M X = 0, where X is some combination of X and X.The other component is gone from the chiral ring.

DISCUSSION
One goal of this program is to search for and study minimal N = 1 SCFTs.One feature of the low-central charge SCFTs we have examined here is that there is a chiral operator satisfying a relation of the form O n ∼ 0 for n = 2, 3. Another feature is that the central charges of the SCFTs considered here lie in a narrow range of a/c.It would be interesting to pursue the reasons for this, and search for other N = 1 SCFTs with truncated chiral rings.
A common property of the RG flows in this landscape is that some operators that are irrelevant at high-energy can be relevant in the IR-such operators are called dangerously irrelevant.As such this is an interesting arena for studying RG flows along the lines of [33].
At present, the status of the 36 "bad" theories is unclear, because it is not clear how to account for the accidental symmetry in the a-maximization procedure and thus check if the corrected theory would flow to an interacting SCFT.One way forward would be to identify the fermionic multiplet that contributes to the unitaryviolating terms in the index and decouple it, as we naively did for the T M theory.It would be interesting to resolve this question and understand how the accidental symmetry arises.This would settle whether one of these theories is indeed a new candidate minimal N = 1 theory, or strengthen the case for minimality of the H * 0 theory.We would like to thank Prarit Agarwal, Matthew Buican, Sergio Benvenuti, Stefano Cremonesi, Thomas Dumitrescu, Ken Intriligator, Kimyeong Lee, Sungjay Lee, Aneesh Manohar, Yu Nakayama, Yuji Tachikawa, and Piljin Yi for helpful discussions.JS would like to thank the UCSD High Energy Theory group for hospitality.The work of KM is supported by JSPS KAKENHI Grant Number JP17K14296.The work of EN was supported in part by DOE grant de-sc0009919, and a UC President's Dissertation Year Fellowship.The work of JS is supported in part by Overseas Research Program for Young Scientists through Korea Institute for Advanced Study (KIAS).Part of this work was performed at the Aspen Center for Physics, which is supported by National Science Foundation grant PHY-1607611.

APPENDIX: List of Candidate SCFTs
Here we list the 70 candidate fixed points under investigation in the main letter.Tables II and IV list the central charges and R-charges of fields for the "good" and "bad" theories, respectively.The "bad" theories are labeled as such when naively the a-maximization procedure holds, but there is a term in the index that indicates a sickness of the theory.We checked terms in the reduced index up to order t 8 to make this determination.We expect that there is something more to understand about these theories, and so have included their data here.
There are generally multiple UV descriptions for the same fixed point.We've checked that these multiple descriptions are consistent with each other, and chosen a representative superpotential that flows to each fixed point.These superpotentials are listed in Tables III and V.
Sometimes the fixed point theory has a flavor symmetry that is not manifest in any of the classical descriptions.This feature can be diagnosed with the t 6 term in the index, whose coefficient counts marginal operators minus conserved flavor currents.When the theory has a non-manifest U (1), we cannot rule out the possibility that it mixes with the superconformal R-symmetry and changes the a-maximization answer.This is a common occurrence in the "bad" theories.We take note of which of the "good" theories have this feature in Table III.(a, c) R(q) R( q) R(φ) R(Xi) R(Mi)

4 .
O } obtained by the deformation δW = O for all O ∈ R T .Consider the fixed points {T O } given by adding an additional gauge-singlet N = 1 chiral field M and the superpotential coupling δW = M O for all O ∈ S T .

FIG. 1 .
FIG.1.The central charges of the 34 "good" theories.The ratios a/c all lie within the range (0.8246, 0.9895).The mean value of a/c is 0.8732 with standard deviation 0.0403.

TABLE I .
The R-charges of the chiral multiplets at various fixed points.The Tµ theory has 3 chiral multiplets labeled M , which we denote as M1,2,3.